# Gradient potential estimates for elliptic double obstacle problems with
Orlicz growth

Qi Xiong  School of Mathematics, Southwest Jiaotong University, Chengdu,
Sichuan, 610031, P.R. China  Zhenqiu Zhang Corresponding author. School of
Mathematical Sciences and LPMC, Nankai University, Tianjin, 300071, P.R. China
Lingwei Ma  School of Mathematical Sciences, Tianjin Normal University,
Tianjin, 300387, P.R. China

###### Abstract

In this paper,we consider the solutions of the elliptic double obstacle
problems with Orlicz growth involving measure data. Some pointwise estimates
for the approximable solutions to these problems are obtained in terms of
fractional maximal operators. Furthermore, we establish pointwise and
oscillation estimates for the gradients of solutions via the non-linear Wolff
potentials, which in turn yield $C^{1,\alpha}$-regularity of solutions.

Mathematics Subject classification (2010): 35B45; 35R05; 35B65.

Keywords: double obstacle problems; Wolff potential estimates; Restricted
fractional maximal functions.

11footnotetext: E-mail: xq@swjtu.edu.cn(Q. Xiong<EMAIL_ADDRESS>(Z.
<EMAIL_ADDRESS>(L. Ma).

##  1 Introduction and main results

We consider the double obstacle problems with Orlicz growth , which are
closely connected to measure data problems of the type

$-\operatorname{div}\left({a}(x,Du)\right)=\mu\quad\quad\mbox{in}\ \ \
\Omega,\\\ $ (1.1)

where $\Omega\subseteq\mathbb{R}^{n},n\geqslant 2$ is a bounded open set and
$\mu\in\mathcal{M}_{b}(\Omega)$, where $\mathcal{M}_{b}(\Omega)$ is the set of
signed Radon measures $\mu$ for which $|\mu|(\Omega)$ is finite and here we
denote by $|\mu|$ the total variation of $\mu$. Moreover we assume that
$\mu(\mathbb{R}^{n}\backslash\Omega)=0$ and
$a=a(x,\eta):\Omega\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ is
measurable for each $x\in\Omega$ and differentiable for almost every
$\eta\in\mathbb{R}^{n}$ and there exist constants $0<l\leqslant 1\leqslant
L<+\infty$ such that for all $x\in\Omega,\eta,\lambda\in\mathbb{R}^{n}$,

$\displaystyle\left\\{\begin{array}[]{r@{}c@{}ll}&&D_{\eta}a(x,\eta)\lambda\cdot\lambda\geqslant
l\dfrac{g(|\eta|)}{|\eta|}|\lambda|^{2}\,,\\\\[1.42271pt]
&&|a(x,\eta)|+|\eta||D_{\eta}a(x,\eta)|\leqslant Lg(|\eta|)\,,\\\\[1.42271pt]
\end{array}\right.$ (1.4)

where $D_{\eta}$ denotes the differentiation in $\eta$ and
$g(t):[0,+\infty)\rightarrow[0,+\infty)$ satisfies

$\displaystyle\left\\{\begin{array}[]{r@{}c@{}ll}&&g(t)=0\ \ \
\Leftrightarrow\ \ \ t=0\,,\\\\[1.42271pt] &&g(\cdot)\in
C^{1}(\mathbb{R}^{+})\,,\\\\[1.42271pt] &&1\leq
i_{g}=:\inf_{t>0}\frac{tg^{\prime}(t)}{g(t)}\leq\sup_{t>0}\frac{tg^{\prime}(t)}{g(t)}=:s_{g}<\infty.\\\\[1.42271pt]
\end{array}\right.$ (1.8)

We define

$G(t):=\int_{0}^{t}g(\tau)\operatorname{d}\\!\tau\ \ \ \mbox{for}\ \ t\geq 0.$
(1.9)

It is straightforward to see that $G(t)$ is convex and strictly increasing.
The standard example for $G(\cdot)$ is

$G(t)=\int_{0}^{t}(\mu+s^{2})^{\frac{p-2}{2}}sds$

with $\mu\geqslant 0,p\geqslant 2$, then (1.4) is reduced to $p$-growth
condition(see [11, 17]).

The double obstacle condition that we impose on the solution is of the form
$\psi_{1}\leqslant u\leqslant\psi_{2}$ a.e. in $\Omega$, where
$\psi_{1},\psi_{2}\in W^{1,G}(\Omega)\cap W^{2,1}(\Omega)$ ) are two given
functions,where $G$ and $W^{1,G}(\Omega)$ are defined as (1.9) and Definition
1.2 respectively. Let $(W^{1,G}(\Omega))^{\prime}$ is the dual of
$W^{1,G}(\Omega)$ and $f\in L^{1}(\Omega)\cap(W^{1,G}(\Omega))^{\prime}$. In
the classical setting, it is natural to consider the double obstacle problem
that can be formulated by the variational inequality

$\int_{\Omega}a(x,Du)\cdot D(v-u)dx\geq\int_{\Omega}f(v-u)dx$ (1.10)

for all functions $v\in u+W_{0}^{1,G}(\Omega)$ that satisfy $\psi_{1}\leqslant
v\leqslant\psi_{2}$ a.e. in $\Omega$. This type of obstacle problem was
motivated by many applications arising from engineering, physics, mechanics
and other fields of applied sciences. However, we are interested in solutions
to the double obstacle problems with measure data in the sense that we want to
replace the inhomogeneity $f$ by a bounded Radon measure $\mu$. These
solutions can be obtained through approximation by using solutions to the
variational inequalities (1.10). The precise definition of approximable
solutions is provided in Definition 1.3.

The construction of the nonlinear operator $a(\cdot,\cdot)$ we discussed above
is based on the model case

$-\operatorname{div}\left(\kappa(x)\frac{g(|Du|)}{|Du|}Du\right)=\mu\quad\quad\mbox{in}\
\ \ \Omega\\\ $

where $\kappa:\Omega\rightarrow[c,+\infty)$ is bounded measurable and
separated from zero function, and $g$ satisfies (1.8). This type of equation
is arising in the fields of fluid dynamics, magnetism, and mechanics, see for
instance [3]. As a class of generalized $p$-Laplacian type elliptic equations,
it was first proposed by Lieberman[26] and moreover, he proved $C^{\alpha}$\-
and $C^{1,\alpha}$-regularity of the solutions for these elliptic equations in
his paper. Since then, substantial advancements have been made in the
regularity theory of such equations. For a comprehensive overview of the topic
we recommend referring to these articles [4, 6, 7, 8, 27, 32, 38, 41, 43].

In this work, our main goal is to derive pointwise and oscillation estimates
for the gradients of solutions to double obstacle problems by the nonlinear
Wolff potentials. The Wolff potential was introduced by Maz’ya and Havin [29]
and the relevant important contributions were attributed to Hedberg and Wolff
[14]. The fundamental results due to Kilpel$\ddot{a}$inen and Mal$\acute{y}$
[15, 16] consist of the pointwise estimates of solutions to the nonlinear
equations of $p$-Laplace type by the Wolff potentials. Later these results
have been extended to a general setting by Trudinger and Wang [35, 36] by
means of a different approach. Further remarkable results for the gradient of
solutions have been achieved by Duzaar, Kuusi and Mingione [10, 11, 17, 28].
Potential estimates are comprehensive tools to obtain regularity estimates.
Indeed, they have been intensively studied and extended in several directions;
we refer to [12, 18, 21] for the elliptic systems, [22, 23] for the parabolic
equations, [33, 34] for single obstacle problems, respectively. For
supplementary results, please see [19, 20, 24, 25, 30, 40].

As for the elliptic equations with Orlicz growth, Baroni [2] obtained
pointwise gradient estimates for solutions of equations with constant
coefficients by the nonlinear potentials. Later, these results were upgraded
by Xiong and Zhang and Ma in [37, 39] to single obstacle problems with measure
data and Orlicz growth. In this paper we consider double obstacle problems. To
overcome the difficulty in this case, we establish a suitable comparison
estimate to transfer the double obstacle problems to the single one, so that
we can still apply the analytic tools in [17, 28]. Initially, we deduce excess
decay estimates for solutions of double obstacle problems through the
utilization of certain comparison estimates. Subsequently, through iterative
procedures based on the obtained estimates, we derive pointwise estimates for
fractional maximal operators. Ultimately, these estimates allow us to draw
conclusions about pointwise and oscillation estimates for the gradients of
solutions.

Next, we summarize our main results. We begin by presenting some definitions,
notations and assumptions.

######  Definition 1.1.

A function $G:[0,+\infty)\rightarrow[0,+\infty)$ is called a Young function if
it is convex and $G(0)=0$.

######  Definition 1.2.

Assume that G is a Young function, the Orlicz class $K^{G}(\Omega)$ is the set
of all measurable functions $u:\Omega\rightarrow\mathbb{R}$ satisfying

$\int_{\Omega}G(|u|)\operatorname{d}\\!\xi<\infty.$

The Orlicz space $L^{G}(\Omega)$ is the linear hull of the Orlicz class
$K^{G}(\Omega)$ with the Luxemburg norm

$\|u\|_{L^{G}(\Omega)}:=\inf\left\\{\alpha>0:\ \
\int_{\Omega}G\left(\frac{|u|}{\alpha}\right)\operatorname{d}\\!\xi\leqslant
1\right\\}.$

Furthermore, the Orlicz-Sobolev space $W^{1,G}(\Omega)$ is defined as

$W^{1,G}(\Omega)=\left\\{u\in L^{G}(\Omega)\cap W^{1,1}(\Omega)\ |\ Du\in
L^{G}(\Omega)\right\\}.$

The space $W^{1,G}(\Omega)$, equipped with the norm
$\|u\|_{W^{1,G}(\Omega)}:=\|u\|_{L^{G}(\Omega)}+\|Du\|_{L^{G}(\Omega)},$ is a
Banach space. Clearly, $W^{1,G}(\Omega)=W^{1,p}(\Omega)$, the standard Sobolev
space, if $G(t)=t^{p}$ with $p\geqslant 1$.

Note that for the Luxemburg norm there holds the inequality

$\|u\|_{L^{G}(\Omega)}\leqslant\int_{\Omega}G(|u|)\operatorname{d}\\!\xi+1.$

The subspace $W_{0}^{1,G}(\Omega)$ is the closure of $C_{0}^{\infty}(\Omega)$
in $W^{1,G}(\Omega)$. The above properties about Orlicz space can be found in
[31].

For every $k>0$ we let

$T_{k}(s):=\left\\{\begin{array}[]{r@{\ \ }c@{\ \ }ll}s\ \ \ \ \ \ \ \ if\ \
|s|\leqslant k\,,\\\\[1.42271pt] k\ sgn(s)\ \ \ \ \ \ \ if\ \
|s|>k\,.\\\\[1.42271pt] \end{array}\right.$

Moreover, for given Dirichlet boundary data $h\in W^{1,G}(\Omega)$, we define

$\mathcal{T}^{1,G}_{h}(\Omega):=\left\\{u:\Omega\rightarrow\mathbb{R}\
measurable:T_{k}(u-h)\in W_{0}^{1,G}(\Omega)\ \ for\ all\ k>0\right\\}.$

We now give the definition of approximable solutions.

######  Definition 1.3.

Suppose that obstacle functions $\psi_{1},\psi_{2}\in W^{1,G}(\Omega)$,
measure data $\mu\in\mathcal{M}_{b}(\Omega)$ and boundary data $h\in
W^{1,G}(\Omega)$ with $\psi_{1}\leqslant h\leqslant\psi_{2}$ a.e. are given.
We say that $u\in\mathcal{T}^{1,G}_{h}(\Omega)$ with $\psi_{1}\leqslant
u\leqslant\psi_{2}$ a.e. on $\Omega$ is a limit of approximating solutions of
the double obstacle problem $OP(\psi_{1},\psi_{2};\mu)$ if there exist
functions

$f_{i}\in(W^{1,G}(\Omega))^{\prime}\cap L^{1}(\Omega)\ \ with\ \
f_{i}\stackrel{{\scriptstyle\ast}}{{\rightharpoonup}}\mu\ in\
\mathcal{M}_{b}(\Omega)\ \ as\ i\rightarrow+\infty$

satisfies

$\limsup_{i\rightarrow+\infty}\int_{B_{R}(x_{0})}|f_{i}|dx\leqslant|\mu|(\overline{B_{R}(x_{0})})$

and solutions $u_{i}\in W^{1,G}(\Omega)$ with $\psi_{1}\leqslant
u_{i}\leqslant\psi_{2}$ of the variational inequalities

$\int_{\Omega}a(x,Du_{i})\cdot
D(v-u_{i})dx\geqslant\int_{\Omega}f_{i}(v-u_{i})dx$ (1.11)

for $\forall\ v\in u_{i}+W_{0}^{1,G}(\Omega)$ with $\psi_{1}\leqslant
v\leqslant\psi_{2}$ a.e. in $\Omega$, such that for $i\rightarrow+\infty$,

$u_{i}\rightarrow u\ \ a.e.\ \ \ in\ \ \Omega$

and

$u_{i}\rightarrow u\ \ \ in\ \ \ W^{1,1}(\Omega).$

The existence of approximating solutions, converging as defined above, has
been proved in our previous work [37] for the inequalities (1.11) with a
single obstacle. Consequently, the existence in this paper can be attained
through minor modifications.

Let us next turn our attention to the classical non-linear Wolff potential
which is defined by

$W^{\mu}_{\beta,p}(x,R):=\int_{0}^{R}\left(\frac{|\mu|(B_{\rho}(x))}{\rho^{n-\beta
p}}\right)^{1/(p-1)}\frac{\operatorname{d}\\!\rho}{\rho}$

for parameters $\beta\in(0,n]$ and $p>1$. We also abbreviate

$W^{[\psi_{1}]}_{\beta,p}(x,R):=\int_{0}^{R}\left(\frac{D\Psi_{1}(B_{\rho}(x))}{\rho^{n-\beta
p}}\right)^{1/(p-1)}\frac{\operatorname{d}\\!\rho}{\rho}$

and

$W^{[\psi_{2}]}_{\beta,p}(x,R):=\int_{0}^{R}\left(\frac{D\Psi_{2}(B_{\rho}(x))}{\rho^{n-\beta
p}}\right)^{1/(p-1)}\frac{\operatorname{d}\\!\rho}{\rho}$

with
$D\Psi_{1}(B_{\rho}(x)):=\int_{B_{\rho}(x)}\left(\frac{g(|D\psi_{1}|)}{|D\psi_{1}|}|D^{2}\psi_{1}|+1\right)d\xi$,
$D\Psi_{2}(B_{\rho}(x)):=\int_{B_{\rho}(x)}|\operatorname{div}a(x,D\psi_{2})|d\xi$.

Now we recall the definitions of the centered maximal operators as follows.

######  Definition 1.4.

Let $\beta\in[0,n],x\in\Omega$ and $R<dist(x,\partial\Omega)$, and let $u$ be
an $L^{1}(\Omega)$-function or a measure with finite mass; the restricted
fractional $\beta$ maximal function of $u$ is defined by

$M_{\beta,R}(u)(x):=\sup_{0<r\leqslant
R}r^{\beta}\frac{|u|(B_{r}(x))}{|B_{r}(x)|}=\sup_{0<r\leqslant
R}r^{\beta}\fint_{B_{r}(x)}|u|\operatorname{d}\\!\xi.$

Note that when $\beta=0$ the one defined above is the classical Hardy-
Littlewood maximal operator.

Moreover, we define

$\overline{M}_{\beta,R}(\psi_{1})(x):=\sup_{0<r\leqslant
R}r^{\beta}\frac{D\Psi_{1}(B_{r}(x))}{|B_{r}(x)|}=\sup_{0<r\leqslant
R}r^{\beta}\fint_{B_{r}(x)}\left(\frac{g(|D\psi_{1}|)}{|D\psi_{1}|}|D^{2}\psi_{1}|+1\right)\operatorname{d}\\!\xi$

and

$\overline{M}_{\beta,R}(\psi_{2})(x):=\sup_{0<r\leqslant
R}r^{\beta}\frac{D\Psi_{2}(B_{r}(x))}{|B_{r}(x)|}=\sup_{0<r\leqslant
R}r^{\beta}\fint_{B_{r}(x)}|\operatorname{div}a(x,D\psi_{2})|\operatorname{d}\\!\xi.$

######  Definition 1.5.

Let $\beta\in[0,n],x\in\Omega$ and $R<dist(x,\partial\Omega)$, and let $u$ be
an $L^{1}(\Omega)$-function or a measure with finite mass; the restricted
sharp fractional $\beta$ maximal function of $u$ is defined by

$M^{\\#}_{\beta,R}(u)(x):=\sup_{0<r\leqslant
R}r^{-\beta}\fint_{B_{r}(x)}|u-(u)_{B_{r}(x)}|\operatorname{d}\\!\xi.$

When $\beta=0$ the one defined above is the Fefferman-Stein sharp maximal
operator.

Throughout this paper we write

$\theta(a,B_{r}(x_{0}))(x):=\sup_{\eta\in\mathbb{R}^{n}\setminus\left\\{0\right\\}}\frac{|a(x,\eta)-\overline{a}_{B_{r}(x_{0})}(\eta)|}{g(|\eta|)},$

where

$\overline{a}_{B_{r}(x_{0})}(\eta):=\fint_{B_{r}(x_{0})}a(x,\eta)dx.$

Then we can easily check from (1.4) that $|\theta(a,B_{r}(x_{0}))|\leqslant
2L$.

######  Definition 1.6.

We say that $a(x,\eta)$ is ($\delta$, R)-vanishing for some $\delta,R>0$, if

$\omega(R):=\sup_{{\begin{subarray}{c}x_{0}\,\in\,\Omega\\\ 0<r\leq
R\end{subarray}}}\fint_{B_{r}(x_{0})}\theta(a,B_{r}(x_{0}))\operatorname{d}\\!x\leq\delta.$
(1.12)

Moreover, ${\omega(\cdot)}^{\frac{1}{1+s_{g}}}$ will be called Dini-BMO
regular if

$\sup_{r>0}\int_{0}^{r}{\omega(\rho)}^{\frac{1}{1+s_{g}}}\frac{d\rho}{\rho}<+\infty;$

${\omega(\cdot)}^{\frac{1}{1+s_{g}}}$ will be called Dini-H$\ddot{o}$lder
regular if

$\sup_{r>0}\int_{0}^{r}\frac{{\omega(\rho)}^{\frac{1}{1+s_{g}}}}{\rho^{\alpha}}\frac{d\rho}{\rho}<+\infty.\
\ \ \ $

Throughout this paper, we always assume that $\delta$ is a small positive
constant.

Finally we state our main results of this paper. The first result we are going
to present is some pointwise estimates of certain maximal operators of
approximable solutions.

######  Theorem 1.7.

Under the assumptions (1.4), (1.8) and (1.12), let $\psi_{1},\psi_{2}\in
W^{1,G}(\Omega)\cap W^{2,1}(\Omega)$,
$\operatorname{div}a(x,D\psi_{2}),\frac{g(|D\psi_{1}|)}{|D\psi_{1}|}|D^{2}\psi_{1}|\in
L^{1}_{loc}(\Omega)$. Assume that $u\in W^{1,1}(\Omega)$ with
$\psi_{1}\leqslant u\leqslant\psi_{2}$ a.e. is a limit of approximating
solutions to $OP(\psi_{1},\psi_{2};\mu)$ with measure data
$\mu\in\mathcal{M}_{b}(\Omega)$(in the sense of Definition 1.3), and
${\omega(\cdot)}^{\frac{1}{1+s_{g}}}$ is Dini-BMO regular in the sense of
Definition 1.6, then there exists a constant $c=c(data)$ and a radius
$R_{0}>0$, depending on $data,\omega(\cdot)$, such that

$\displaystyle M_{\alpha,R}^{\\#}(u)(x)+M_{1-\alpha,R}(Du)(x)$ (1.13)
$\displaystyle\leqslant$ $\displaystyle
c\left[W_{1-\alpha+\frac{\alpha}{i_{g}+1},i_{g}+1}^{\mu}(x,2R)+W_{1-\alpha+\frac{\alpha}{i_{g}+1},i_{g}+1}^{[\psi_{1}]}(x,2R)+W_{1-\alpha+\frac{\alpha}{i_{g}+1},i_{g}+1}^{[\psi_{2}]}(x,2R)\right]$
$\displaystyle+$ $\displaystyle
cR^{1-\alpha}\fint_{B_{R}(x)}|Du|d\xi+c\int_{0}^{2R}[\omega(\rho)]^{\frac{1}{1+s_{g}}}G^{-1}\left[\fint_{B_{\rho}(x)}[G(|D\psi_{1}|)+G(|\psi_{1}|)]d\xi\right]\frac{d\rho}{\rho^{\alpha}}.$

Further assume that

$\sup_{r>0}\frac{[\omega(r)]^{\frac{1}{1+s_{g}}}}{r^{\widehat{\alpha}}}\leqslant
c_{0},$ (1.14)

for some $\widehat{\alpha}\in[0,\beta)$, then

$\displaystyle M_{\alpha,R}^{\\#}(Du)(x)$ (1.15) $\displaystyle\leqslant$
$\displaystyle c\left\\{\left[M_{1-\alpha
i_{g},R}(\mu)(x)\right]^{\frac{1}{i_{g}}}+\left[\overline{M}_{1-\alpha
i_{g},R}(\psi_{1})(x)\right]^{\frac{1}{i_{g}}}+\left[\overline{M}_{1-\alpha
i_{g},R}(\psi_{2})(x)\right]^{\frac{1}{i_{g}}}\right\\}$ $\displaystyle+$
$\displaystyle
c\left[W_{\frac{1}{i_{g}+1},i_{g}+1}^{\mu}(x,2R)+W_{\frac{1}{i_{g}+1},i_{g}+1}^{[\psi_{1}]}(x,2R)+W_{\frac{1}{i_{g}+1},i_{g}+1}^{[\psi_{2}]}(x,2R)\right]$
$\displaystyle+$ $\displaystyle
cR^{-\alpha}\fint_{B_{R}(x)}|Du|\operatorname{d}\\!\xi+c\int_{0}^{2R}[\omega(\rho)]^{\frac{1}{1+s_{g}}}G^{-1}\left[\fint_{B_{\rho}(x)}[G(|D\psi_{1}|)+G(|\psi_{1}|)]d\xi\right]\frac{d\rho}{\rho^{1+\alpha}}$

holds uniformly in $\alpha\in[0,\widehat{\alpha}]$, where
$c=c(data,\widehat{\alpha},c_{0},\omega(\cdot),diam(\Omega))$,
$0<R\leqslant\min\left\\{R_{0},dist(x_{0},\partial\Omega)\right\\}$ and
$\beta$ is as in Lemma 3.7.

The previous theorem implies the pointwise and oscillation estimates for the
gradients of solutions to double obstacle problems.

######  Theorem 1.8.

In the same hypotheses of Theorem 1.7, let
$B_{4R}(x_{0})\subseteq\Omega,x,y\in
B_{\frac{R}{4}}(x_{0}),0<R\leqslant\frac{1}{2},$ and for some
$\widehat{\alpha}\in[0,\beta)$, assume that
${\omega(\cdot)}^{\frac{1}{1+s_{g}}}$ is Dini-H$\ddot{o}$lder regular, that is

$c_{0}:=\sup_{r>0}\int_{0}^{r}\frac{[\omega(\rho)]^{\frac{1}{1+s_{g}}}}{\rho^{\widehat{\alpha}}}\frac{d\rho}{\rho}<+\infty.$
(1.16)

Then there exists a constant
$c=c(n,i_{g},s_{g},v,L,c_{0},\widehat{\alpha},\omega(\cdot),diam(\Omega))$
such that

$\displaystyle|Du(x_{0})|$ (1.17) $\displaystyle\leqslant$ $\displaystyle
c\left[\fint_{B_{R}(x_{0})}|Du|d\xi+W_{\frac{1}{i_{g}+1},i_{g}+1}^{\mu}(x_{0},2R)+W_{\frac{1}{i_{g}+1},i_{g}+1}^{[\psi_{1}]}(x_{0},2R)+W_{\frac{1}{i_{g}+1},i_{g}+1}^{[\psi_{2}]}(x_{0},2R)\right]$
$\displaystyle+$ $\displaystyle
c\int_{0}^{2R}[\omega(\rho)]^{\frac{1}{1+s_{g}}}G^{-1}\left[\fint_{B_{\rho}(x_{0})}[G(|D\psi_{1}|)+G(|\psi_{1}|)]d\xi\right]\frac{d\rho}{\rho},$
$\displaystyle|Du(x)-Du(y)|$ (1.18) $\displaystyle\leq$ $\displaystyle
c\fint_{B_{R}(x_{0})}|Du|\operatorname{d}\\!\xi\left(\frac{|x-y|}{R}\right)^{\alpha}$
$\displaystyle+$ $\displaystyle
c\left[W^{\mu}_{-\alpha+\frac{1+\alpha}{1+i_{g}},i_{g}+1}(x,2R)+W^{[\psi_{1}]}_{-\alpha+\frac{1+\alpha}{1+i_{g}},i_{g}+1}(x,2R)+W^{[\psi_{2}]}_{-\alpha+\frac{1+\alpha}{1+i_{g}},i_{g}+1}(x,2R)\right]|x-y|^{\alpha}$
$\displaystyle+$ $\displaystyle
c\left[W^{\mu}_{-\alpha+\frac{1+\alpha}{1+i_{g}},i_{g}+1}(y,2R)+W^{[\psi_{1}]}_{-\alpha+\frac{1+\alpha}{1+i_{g}},i_{g}+1}(y,2R)++W^{[\psi_{2}]}_{-\alpha+\frac{1+\alpha}{1+i_{g}},i_{g}+1}(y,2R)\right]|x-y|^{\alpha}$
$\displaystyle+$ $\displaystyle
c\left[\int_{0}^{2R}[\omega(\rho)]^{\frac{1}{1+s_{g}}}G^{-1}\left[\fint_{B_{\rho}(x)}[G(|D\psi_{1}|)+G(|\psi_{1}|)]d\xi\right]\frac{d\rho}{\rho^{1+\alpha}}\right]|x-y|^{\alpha}$
$\displaystyle+$ $\displaystyle
c\left[\int_{0}^{2R}[\omega(\rho)]^{\frac{1}{1+s_{g}}}G^{-1}\left[\fint_{B_{\rho}(y)}[G(|D\psi_{1}|)+G(|\psi_{1}|)]d\xi\right]\frac{d\rho}{\rho^{1+\alpha}}\right]|x-y|^{\alpha},$

holds uniformly in $\alpha\in[0,\widehat{\alpha}]$, where $\beta$ is as in
Lemma 3.7 and $x,y$ is the Lebesgue’s point of $Du$.

######  Remark 1.9.

To the best of our knowledge, there is not much research on the gradient
estimate associated with the double obstacle problems, and our work is new,
and our results provide a new perspective for understanding the solutions to
the double obstacle problems.

The remainder of this paper is organized as follows. Section 2 contains some
notions and preliminary results. In Section 3, we derive the excess decay
estimate for solutions to these problems by using some comparison estimates.
In Section 4, we obtain pointwise and oscillation estimates for the gradients
of solutions by the Wolff potentials.

##  2 Preliminaries

In this section, we introduce some notions and results which will be used in
this paper. For brevity in notation, we gather the dependencies of specific
constants on the parameters of our inquiry as

$data=data(n,i_{g},s_{g},l,L).$

For an integrable map $f:\Omega\rightarrow\mathbb{R}^{n}$, we write

$(f)_{\Omega}:=\fint_{\Omega}fdx:=\frac{1}{|\Omega|}\int_{\Omega}fdx.$

For $q\in[1,\infty)$, it is easily verified that

$\parallel f-(f)_{\Omega}\parallel_{L^{q}(\Omega)}\leqslant
2\min_{c\in\mathbb{R}^{m}}\parallel f-c\parallel_{L^{q}(\Omega)}.$ (2.1)

######  Definition 2.1.

A Young function $G$ is called an $N$-function if

$0<G(t)<+\infty\ \ for\ t>0$

and

$\lim_{t\rightarrow+\infty}\frac{G(t)}{t}=\lim_{t\rightarrow
0}\frac{t}{G(t)}=+\infty.$ (2.2)

It’s obvious that $G(t)$ is an $N$-function.

The Young conjugate of a Young function G will be denoted by $G^{\ast}$ and
defined as

$G^{\ast}(t)=\sup_{s\geq 0}\left\\{st-G(s)\right\\}\ \ for\ t\geq 0.$

In particular, if $G$ is an $N$-function, then $G^{\ast}$ is an $N$-function
as well.

######  Definition 2.2.

A Young function G is said to satisfy the global $\vartriangle_{2}$ condition,
denoted by $G\in\vartriangle_{2}$, if there exists a positive constant C such
that for every $t>0$,

$G(2t)\leq CG(t).$

Similarly, a Young function G is said to satisfy the global
$\bigtriangledown_{2}$ condition, denoted by $G\in\bigtriangledown_{2}$, if
there exists a constant $\theta>1$ such that for every $t>0$,

$G(t)\leq\frac{G(\theta t)}{2\theta}.$

######  Remark 2.3.

For an increasing function $f:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$
satisfying $\vartriangle_{2}$ condition $f(2t)\lesssim f(t)$ for $t\geqslant
0$, it is easy to prove that $f(t+s)\leqslant c[f(t)+f(s)]$ holds for every
$t,s\geqslant 0$.

Next let us recall a basic property of $N$-function, which will be used in the
sequel.

######  Lemma 2.4.

[42] If $G$ is an $N$-function, then $G$ satisfies the following Young’s
inequality

$st\leq G^{*}(s)+G(t),\ \ \ for\ \ \forall s,t\geq 0.$

Furthermore, if $G\in\bigtriangleup_{2}\cap\bigtriangledown_{2}$ is an
$N$-function, then $G$ satisfies the following Young’s inequality with
$\forall\varepsilon>0$,

$st\leq\varepsilon G^{*}(s)+c(\varepsilon)G(t),\ \ \ for\ \ \forall s,t\geq
0.$

Another important property of Young’s conjugate function is the following
inequality, which can be found in [1]:

$G^{*}\left(\frac{G(t)}{t}\right)\leqslant G(t).$ (2.3)

######  Lemma 2.5.

[7, 42] Under the assumption (1.8), G(t) is defined in (1.9). Then we have

(1) $G(t)$ is strictly convex $N$-function and

$G^{\ast}(g(t))\leqslant cG(t)\ \ \ for\ t\geqslant 0\ \ \ and\ \ some\ \
c>0;$

(2) $G(t)\in\bigtriangledown_{2}.$

######  Lemma 2.6.

[5, 9] Under the assumptions (1.4) and (1.8), G(t) is defined in (1.9). Then
there exists $c=c(data)>0$ such that

$[a(x,\eta)-a(x,\xi)]\cdot(\eta-\xi)\geq cG(|\eta-\xi|),\ \ \ \ for\ \ every\
\ x\in\Omega,\eta,\xi,\in\mathbb{R}^{n}.$ (2.4)

Especially, we have

$a(x,\eta)\cdot\eta\geq cG(|\eta|),\ \ \ \ \ for\ \ every\ \
x\in\Omega,\eta\in\mathbb{R}^{n}.$ (2.5)

The following iteration lemma turns out to be very useful in the sequel.

######  Lemma 2.7.

[13] Let $f(t)$ be a nonnegative function defined on the interval $[a,b]$ with
$a\geqslant 0$. Suppose that for $s,t\in[a,b]$ with $t<s$,

$f(t)\leqslant\frac{A}{(s-t)^{\alpha}}+\frac{B}{(s-t)^{\beta}}+C+\theta f(s)$

holds, where $A,B,C\geqslant 0,\alpha,\beta>0$ and $0\leqslant\theta<1$. Then
there exists a constant $c=c(\alpha,\theta)$ such that

$f(\rho)\leqslant
c\left(\frac{A}{(R-\rho)^{\alpha}}+\frac{B}{(R-\rho)^{\beta}}+C\right)$

for any $\rho,R\in[a,b]$ with $\rho<R$.

The proof of the following lemma can be found in [37].

######  Lemma 2.8.

Let $\Omega\subset\mathbb{R}$ be a bounded domain. Assume that
$1+i_{g}\leqslant n$ , $h\in\mathcal{T}^{1,G}_{0}(\Omega)$ satisfies

$\int_{\Omega\cap\left\\{|h|\leqslant k\right\\}}|Dh|^{1+i_{g}}dx\leqslant
Mk+M^{\frac{1+i_{g}}{i_{g}}}$

for $\forall\ k>0$, and fixed constants $M>0$. Then we have

$\int_{\Omega}|h|^{1+\alpha}dx\leqslant c_{1}M^{\frac{1+\alpha}{i_{g}}},$
$\int_{\Omega}|Dh|^{1+\beta}dx\leqslant c_{2}M^{\frac{1+\beta}{i_{g}}},$

where
$0<\alpha<min\left\\{1,\frac{n(i_{g}-1)+1+i_{g}}{n-1-i_{g}}\right\\},0<\beta<min\left\\{1,\frac{n(i_{g}-1)+1}{n-1}\right\\},c_{1}=c_{1}(\Omega,n,i_{g},\alpha),$
$c_{2}=c_{2}(\Omega,n,i_{g},\beta).$

######  Lemma 2.9.

Assume that $g(t)$ satisfies (1.8), $G(t)$ is defined in (1.9). Then we have

(1) for any $\beta\geq 1$,

$\beta^{i_{g}}\leq\dfrac{g(\beta t)}{g(t)}\leq\beta^{s_{g}}$ and
$\beta^{1+i_{g}}\leq\dfrac{G(\beta t)}{G(t)}\leq\beta^{1+s_{g}}$, for every
$t>0$,

for any $0<\beta<1$,

$\beta^{s_{g}}\leq\dfrac{g(\beta t)}{g(t)}\leq\beta^{i_{g}}$ and
$\beta^{1+s_{g}}\leq\dfrac{G(\beta t)}{G(t)}\leq\beta^{1+i_{g}}$, for every
$t>0$.

(2) for any $\beta\geq 1$,

$\beta^{\frac{1}{s_{g}}}\leq\dfrac{g^{-1}(\beta
t)}{g^{-1}(t)}\leq\beta^{\frac{1}{i_{g}}}$ and
$\beta^{\frac{1}{1+s_{g}}}\leq\dfrac{G^{-1}(\beta
t)}{G^{-1}(t)}\leq\beta^{\frac{1}{1+i_{g}}}$, for every $t>0$,

for any $0<\beta<1$,

$\beta^{\frac{1}{i_{g}}}\leq\dfrac{g^{-1}(\beta
t)}{g^{-1}(t)}\leq\beta^{\frac{1}{s_{g}}}$ and
$\beta^{\frac{1}{1+i_{g}}}\leq\dfrac{G^{-1}(\beta
t)}{G^{-1}(t)}\leq\beta^{\frac{1}{1+s_{g}}}$, for every $t>0$.

It’s obvious that Lemma 2.9 implies that

$L^{1+s_{g}}(\Omega)\subset L^{G}(\Omega)\subset L^{1+i_{g}}(\Omega)\subset
L^{1}(\Omega),$ (2.6)

and $g$ and $G$ satisfy $\vartriangle_{2}$ condition. Then from Remark 2.3, we
know that $g$ and $G$ satisfy the subadditivity property: $g(t+s)\leqslant
c[g(t)+g(s)]$, $G(t+s)\leqslant c[G(t)+G(s)],$ for every $t,s\geqslant 0$.

##  3 Comparison estimates

This section is devoted to comparing the solutions of double obstacle problems
to the solutions of elliptic equations. Therefore we can obtain a similar
excess decay estimate for solutions of double obstacle problems with measure
data. Firstly, we shall prove the comparison estimate between the solutions of
double obstacle problems with measure data and the solutions of the single
obstacle problems.

######  Lemma 3.1.

Assume that conditions (1.4)-(1.8) are fulfilled, let
$B_{2R}(x_{0})\subset\Omega,f\in
L^{1}(B_{R}(x_{0}))\cap(W^{1,G}(B_{R}(x_{0})))^{\prime}$ and the map $u\in
W^{1,G}(B_{R}(x_{0}))$ with $\psi_{1}\leqslant u\leqslant\psi_{2}$ solves the
variational inequality

$\int_{B_{R}(x_{0})}a(x,Du)\cdot D(v-u)dx\geq\int_{B_{R}(x_{0})}f(v-u)dx$
(3.1)

for any $v\in u+W_{0}^{1,G}(B_{R}(x_{0}))$ that satisfy $\psi_{1}\leqslant
v\leqslant\psi_{2}$ a.e. in $B_{R}(x_{0})$. Let $w_{1}\in
u+W_{0}^{1,G}(B_{R}(x_{0}))$ with $w_{1}\geq\psi_{1}$ be the weak solution of
the single obstacle problem

$\int_{B_{R}(x_{0})}a(x,Dw_{1})\cdot
D(v-w_{1})dx\geq\int_{B_{R}(x_{0})}a(x,D\psi_{2})\cdot D(v-w_{1})dx$ (3.2)

for any $v\in w_{1}+W_{0}^{1,G}(B_{R}(x_{0}))$ that satisfy
$v\geqslant\psi_{1}$ a.e. in $B_{R}(x_{0})$. Then there exists $c=c(data)$
such that

$\fint_{B_{R}(x_{0})}|Du-Dw_{1}|\operatorname{d}x\leqslant
c\left[R\fint_{B_{R}(x_{0})}|f|+|\operatorname{div}a(x,D\psi_{2})|dx\right]^{\frac{1}{i_{g}}}.$
(3.3)

###### Proof.

Without loss of generality we may assume that $x_{0}=0,R=1$ by defining

$\widehat{u}(x)=\dfrac{u(Rx+x_{0})}{R},\ \ \
\widehat{w_{1}}(x)=\dfrac{w_{1}(Rx+x_{0})}{R},\ \ \
\widehat{f}(x)=Rf(Rx+x_{0}),\ \ \ $
$\widehat{\psi_{2}}(x)=\dfrac{\psi_{2}(Rx+x_{0})}{R},\ \ \
\widehat{a}(a,\eta)=a(Rx+x_{0},\eta).$

Since $w_{1}\in u+W_{0}^{1,G}(B_{1}),u\leqslant\psi_{2}$ a.e. in $B_{1}$, we
have $(w_{1}-\psi_{2})_{+}\in W_{0}^{1,G}(B_{1})$. Then we take
$v=\min\\{w_{1},\psi_{2}\\}=w_{1}-(w_{1}-\psi_{2})_{+}\in
w_{1}+W_{0}^{1,G}(B_{1})$ with $v\geqslant\psi_{1}$ as comparison functions in
(3.2), it follows from Lemma 2.6 that

$\displaystyle\int_{B_{1}}G(|D(w_{1}-\psi_{2})_{+}|)dx$
$\displaystyle\leqslant$ $\displaystyle
c\int_{B_{1}}[a(x,Dw_{1})-a(x,D\psi_{2})]\cdot D[(w_{1}-\psi_{2})_{+}]dx$
$\displaystyle\leqslant$ $\displaystyle 0.$

Because $G(\cdot)$ is increasing over $[0,+\infty)$ and $G(0)=0$, we can infer
that

$D(w_{1}-\psi_{2})_{+}=0\ \ a.e.\ \ \ in\ \ B_{1}.$

Combining with $(w_{1}-\psi_{2})_{+}=0\ \ \ on\ \ \partial B_{1}$, which
implies

$(w_{1}-\psi_{2})_{+}=0\ \ \ a.e.\ \ in\ \ B_{1}.$

It means that $w_{1}\leqslant\psi_{2}$ a.e. in $B_{1}$.

Case 1: $\int_{B_{1}}|f|+|\operatorname{div}a(x,D\psi_{2})|dx\leqslant 1$. If
$1+i_{g}>n$, because of $u-w_{1}\in W_{0}^{1,G}(B_{1})$, $u-w_{1}\in
W_{0}^{1,1+i_{g}}(B_{1})$, then we make use of Sobolev’s inequality to get
$u-w_{1}\in L^{\infty}(B_{1})$. Now we take $v=\frac{u+w_{1}}{2}\in
u+W_{0}^{1,G}(B_{1})$ as comparison functions in the variational inequalities
(3.1) and (3.2), which implies that

$\displaystyle\int_{B_{1}}|Du-Dw_{1}|^{1+i_{g}}dx$ $\displaystyle\leq$
$\displaystyle c\int_{B_{1}}[G(|Du-Dw_{1}|)+1]dx$ $\displaystyle\leq$
$\displaystyle c\int_{B_{1}}[a(x,Dw_{1})-a(x,Du)]\cdot(Dw_{1}-Du)dx+c$
$\displaystyle\leq$ $\displaystyle
c\int_{B_{1}}\left[|f|+|\operatorname{div}a(x,D\psi_{2})|\right]|w_{1}-u|dx+c$
$\displaystyle\leq$ $\displaystyle c\parallel
u-w_{1}\parallel_{L^{\infty}(B_{1})}\int_{B_{1}}|f|+|\operatorname{div}a(x,D\psi_{2})|dx+c$
$\displaystyle\leq$ $\displaystyle c\parallel Du-
Dw_{1}\parallel_{L^{1+i_{g}}(B_{1})}+c,$

where we used Lemma 2.9 and Lemma 2.6. Then we have

$\int_{B_{1}}|Du-Dw_{1}|dx\leqslant c.$

If $1+i_{g}\leqslant n$, we define

$D_{k}:=\left\\{x\in B_{1}:|u(x)-w_{1}(x)|\leq k\right\\},\ \ \ \forall\ k>0,$

and let $v_{k}:=u+T_{k}(w_{1}-u)$ and
$\overline{v_{k}}:=w_{1}-T_{k}(w_{1}-u)\in u+W_{0}^{1,G}(B_{1})$ as comparison
functions in the variational inequalities (3.1) and (3.2) respectively, then
we have

$\int_{B_{1}}[a(x,Dw_{1})-a(x,Du)]\cdot D[T_{k}(w_{1}-u)]dx\leq
k\int_{B_{1}}|f|+|\operatorname{div}a(x,D\psi_{2})|dx.$ (3.4)

Then for every $k\geqslant 1$, we have

$\displaystyle\int_{D_{k}}|Du-Dw_{1}|^{1+i_{g}}dx$ $\displaystyle\leq$
$\displaystyle c\int_{B_{1}}[a(x,Dw_{1})-a(x,Du)]\cdot D[T_{k}(w_{1}-u)]dx+c$
$\displaystyle\leq$ $\displaystyle ck.$

Now using Lemma 2.8, we obtain

$\int_{B_{1}}|Du-Dw_{1}|dx\leqslant c.$

Case 2: $\int_{B_{1}}|f|+|\operatorname{div}a(x,D\psi_{2})|dx>1$. We let

$\overline{u}(x)=A^{-1}u(x),\ \ \ \overline{v}(x)=A^{-1}v(x),\ \ \
\overline{w_{1}}(x)=A^{-1}w_{1}(x),\ \ \
\overline{\psi_{1}}(x)=A^{-1}\psi_{1}(x)$
$\overline{\psi_{2}}(x)=A^{-1}\psi_{2}(x),\ \ \
\overline{f}(x)=A^{-i_{g}}f(x),\ \ \
\overline{a}(x,\eta)=A^{-i_{g}}a(x,A\eta),\ \ \
\overline{G}(t)=\int_{0}^{t}\overline{g}(\tau)d\tau,$

where
$A=(\int_{B_{1}}|f|+|\operatorname{div}a(x,D\psi_{2})|dx)^{\frac{1}{i_{g}}}>1$.
Then we can easily obtain
$\int_{B_{1}}|\overline{f}|+|\operatorname{div}a(x,D\psi_{2})|dx\leqslant 1$,
and $\overline{a}$ satisfies (1.4) and $\overline{g}$ satisfies (1.8).

Moreover, $\overline{u}\in W^{1,G}(B_{1})$ with
$\overline{\psi_{1}}\leqslant\overline{u}\leqslant\overline{\psi_{2}}$ solves
the variational inequality

$\int_{B_{1}}\overline{a}(x,D\overline{u})\cdot
D(\overline{v}-\overline{u})dx\geq\int_{B_{1}}(\overline{v}-\overline{u})\overline{f}dx$

for any $\overline{v}\in\overline{u}+W_{0}^{1,G}(B_{1})$ that satisfy
$\overline{\psi_{1}}\leqslant\overline{v}\leqslant\overline{\psi_{2}}$ a.e. on
$B_{1}$.

And $\overline{w_{1}}\in\overline{u}+W_{0}^{1,G}(B_{1})$ with
$\overline{w_{1}}\geq\overline{\psi_{1}}$ solves the inequality

$\int_{B_{1}}\overline{a}(x,D\overline{w_{1}})\cdot
D(\overline{v}-\overline{w_{1}})dx\geq\int_{B_{1}}\overline{a}(x,D\overline{\psi_{2}})\cdot
D(\overline{v}-\overline{w_{1}})dx.$

Similar to the case 1 we have

$\int_{B_{1}}|D\overline{u}-D\overline{w_{1}}|dx\leqslant c.$

In conclusion, we have

$\int_{B_{1}}|Du-Dw_{1}|dx\leqslant
c\left(\int_{B_{1}}|f|+|\operatorname{div}a(x,D\psi_{2})|dx\right)^{\frac{1}{i_{g}}},$

which finishes our proof. ∎

######  Corollary 3.2.

Assume that conditions (1.4)-(1.8) are fulfilled, let $w_{1}$ be as in Lemma
3.1 and $\mu\in\mathcal{M}_{b}(\Omega)$ and u be a limit of approximating
solutions for $OP(\psi_{1},\psi_{2};\mu)$, in the sense of Definition 1.3.
Then there exists $c=c(data)$ such that

$\fint_{B_{R}(x_{0})}|Du-Dw_{1}|dx\leqslant
c\left[\frac{|\mu|(\overline{B_{R}}(x_{0}))}{R^{n-1}}+R\fint_{B_{R}(x_{0})}|\operatorname{div}a(x,D\psi_{2})|dx\right]^{\frac{1}{i_{g}}}.$
(3.5)

The proof of corollary 3.2 follows a similar approach to the proof of
corollary 4.2 presented in [37]; hence, we omit its repetition in this
exposition.

The following results have been proved in our previous work [37, 39].

######  Lemma 3.3.

Assume that conditions (1.4)-(1.8) are fulfilled, let
$B_{2R}(x_{0})\subset\Omega,f\in
L^{1}(B_{R}(x_{0}))\cap(W^{1,G}(B_{R}(x_{0})))^{\prime}$ and the map $w_{1}\in
W^{1,G}(B_{R}(x_{0}))$ with $w_{1}\geq\psi_{1}$ solves the variational
inequality (3.2). Let $w_{2}\in w_{1}+W_{0}^{1,G}(B_{R}(x_{0}))$ with
$w_{2}\geq\psi_{1}$ be the weak solution of the homogeneous obstacle problem

$\int_{B_{R}(x_{0})}a(x,Dw_{2})\cdot D(v-w_{2})dx\geqslant 0$ (3.6)

for any $v\in w_{2}+W_{0}^{1,G}(B_{R}(x_{0}))$ with $v\geqslant\psi_{1}\ a.e.\
in\ B_{R}(x_{0})$. Then there exists $c=c(data)$ such that

$\fint_{B_{R}(x_{0})}|Dw_{1}-Dw_{2}|\operatorname{d}x\leqslant
c\left[R\fint_{B_{R}(x_{0})}|\operatorname{div}a(x,D\psi_{2})|dx\right]^{\frac{1}{i_{g}}}.$

######  Lemma 3.4.

Under the conditions (1.4)-(1.8), let $B_{2R}(x_{0})\subseteq\Omega$,
$w_{2}\geqslant 0$, $\psi_{1}$ $\in W^{1,G}(B_{2R}(x_{0}))$, $w_{2}\in
W^{1,G}(B_{2R}(x_{0}))$ with $w_{2}\geqslant\psi_{1}$ solve the inequality
(3.6) and $w_{3}\in W^{1,G}(B_{2R}(x_{0}))$ with $w_{3}\geqslant\psi_{1}$
solve the inequality

$\int_{B_{R}(x_{0})}\overline{a}_{B_{R}(x_{0})}(Dw_{3})\cdot
D(v-w_{3})dx\geqslant 0$ (3.7)

and $w_{3}=w_{2}$ on $\partial B_{R}(x_{0})$. Then we have

$\displaystyle\fint_{B_{R}(x_{0})}|Dw_{2}-Dw_{3}|dx$ $\displaystyle\leqslant$
$\displaystyle
c\omega(R)^{\frac{1}{1+s_{g}}}\left\\{\fint_{B_{2R}(x_{0})}|Dw_{2}|dx+G^{-1}\left[\fint_{B_{2R}(x_{0})}[G(|D\psi_{1}|)+G(|\psi_{1}|)]dx\right]\right\\},$

where $c=c(data)$.

######  Lemma 3.5.

Assume that conditions (1.4)-(1.8) are fulfilled, and $z\in
W^{1,G}(B_{R}(x_{0}))$ with $z\geqslant\psi_{1}$ solves the inequality (3.6).
Let $w\in z+W_{0}^{1,G}(B_{R}(x_{0}))$ be a weak solution of the equation

$-\operatorname{div}\left(a(x,Dw)\right)=-\operatorname{div}\left(a(x,D\psi_{1})\right)\
\ \ \ in\ \ B_{R}(x_{0})$

and $\psi_{1}\in W^{1,G}(B_{R}(x_{0}))\cap W^{2,1}(B_{R}(x_{0})),\
\operatorname{div}\left(a(x,D\psi_{1})\right)\in L^{1}(B_{R}(x_{0})).$ Then
there exists $c=c(data)>0$ such that

$\fint_{B_{R}(x_{0})}|Dz-Dw|dx\leqslant
c\left(R\fint_{B_{R}(x_{0})}(|\operatorname{div}\left(a(x,D\psi_{1})\right)|+1)dx\right)^{\frac{1}{i_{g}}}.$
(3.8)

######  Lemma 3.6.

Assume that conditions (1.4)-(1.8) are fulfilled, let $f,g\in
L^{1}(B_{R}(x_{0}))\cap(W^{1,G}(B_{R}(x_{0})))^{\prime}$ and $z,w\in
W^{1,G}(B_{R}(x_{0}))$ with $z-w\in W_{0}^{1,G}(B_{R}(x_{0}))$ be weak
solutions of

$\left\\{\begin{array}[]{r@{\ \ }c@{\ \
}ll}-\operatorname{div}\left(a(x,Dz)\right)=f&\mbox{in}\ \
B_{R}(x_{0})\,,\\\\[1.42271pt]
-\operatorname{div}\left(a(x,Dw)\right)=g&\mbox{in}\ \
B_{R}(x_{0})\,.\\\\[1.42271pt] \end{array}\right.$ (3.9)

Then the following comparison estimates hold:

$\fint_{B_{R}(x_{0})}|Dz-Dw|dx\leqslant
c\left(R\fint_{B_{R}(x_{0})}(|f|+|g|+1)dx\right)^{\frac{1}{i_{g}}},$ (3.10)

where $c=c(data)>0$.

Next we state an excess decay estimate for a homogeneous comparison problem.

######  Lemma 3.7.

(see [2], lemma 4.1) If $z\in W_{loc}^{1,G}(\Omega)$ is a local weak solution
of

$-\operatorname{div}\left(a(Dz)\right)=0\ \ \ \ \ \ in\ \ \ \Omega,$ (3.11)

under the assumptions (1.4) and (1.8), then there exist constants
$\beta\in(0,1)$ and $c=c(data)$ such that

$\fint_{B_{\rho}(x_{0})}|Dz-(Dz)_{B_{\rho}(x_{0})}|\operatorname{d}\\!\xi\leq
c\left(\frac{\rho}{R}\right)^{\beta}\fint_{B_{R}(x_{0})}|Dz-(Dz)_{B_{R}(x_{0})}|\operatorname{d}\\!\xi,$
(3.12)

where $0<\rho\leqslant R,\ B_{2R}(x_{0})\subset\Omega.$

Our aim is to establish a similar excess decay estimate, incorporating error
terms, for solutions to (3.1). To accomplish this, we utilize a multistep
comparison argument to transfer the excess decay estimate from Lemma 3.7 onto
solutions of the double obstacle problem.

######  Lemma 3.8.

Assume that conditions (1.4)-(1.8) are fulfilled, let
$B_{2R}(x_{0})\subset\Omega,\psi_{1},\psi_{2}\in W^{1,G}(\Omega)\cap
W^{2,1}(\Omega)$,
$\operatorname{div}a(x,D\psi_{2}),\frac{g(|D\psi_{1}|)}{|D\psi_{1}|}|D^{2}\psi_{1}|\in
L^{1}_{loc}(\Omega),$ Let $u\in W^{1,1}(B_{R}(x_{0}))$ with
$u\geqslant\psi_{1}$ a.e. be a limit of approximating solutions for
$OP(\psi_{1},\psi_{2};\mu)$ with measure data
$\mu\in\mathcal{M}_{b}(B_{R}(x_{0}))$ (in the sense of Definition 1.3). Then
there exists $\beta\in(0,1)$ such that

$\displaystyle\fint_{B_{\rho}(x_{0})}|Du-(Du)_{B_{\rho}(x_{0})}|dx\leqslant
c\left(\frac{\rho}{R}\right)^{\beta}\fint_{B_{R}(x_{0})}|Du-(Du)_{B_{R}(x_{0})}|dx$
$\displaystyle+$ $\displaystyle
c\left(\frac{R}{\rho}\right)^{n}\left[\frac{|\mu|(\overline{B_{R}(x_{0})})}{R^{n-1}}+R\fint_{B_{R}}|\operatorname{div}a(x,D\psi_{2})|dx\right]^{\frac{1}{i_{g}}}$
$\displaystyle+$ $\displaystyle
c\left(\frac{R}{\rho}\right)^{n}\left[R\fint_{B_{R}(x_{0})}\left(\frac{g(|D\psi_{1}|)}{|D\psi_{1}|}|D^{2}\psi_{1}|+1\right)dx\right]^{\frac{1}{i_{g}}}$
$\displaystyle+$ $\displaystyle
c\left(\frac{R}{\rho}\right)^{n}\omega(R)^{\frac{1}{1+s_{g}}}\left\\{\fint_{B_{R(x_{0})}}|Du|dx+G^{-1}\left[\fint_{B_{R(x_{0})}}[G(|D\psi_{1}|)+G(|\psi_{1}|)]dx\right]\right\\},$

where $0<\rho\leqslant R,\ c=c(data)$ and $\beta$ is as in Lemma 3.7.

######  Remark 3.9.

As we have successfully derived all the requisite comparison estimates for
regularized problems at the $L^{1}-$level, our results seamlessly extend to
problems involving measure data.

###### Proof.

Without loss of generality we may assume that $x_{0}=0$ and that
$w_{1},w_{2},w_{3},w_{4},w_{5}\in W^{1,G}(B_{R})$ satisfy separately

$\left\\{\begin{array}[]{r@{\ \ }c@{\ \ }ll}&\int_{B_{R}}&a(x,Dw_{1})\cdot
D(v-w_{1})dx\geqslant\int_{B_{R}}a(x,D\psi_{2})\cdot
D(v-w_{1})dx\,,\\\\[1.42271pt] &\mbox{for}&\forall\ v\in
w_{1}+W_{0}^{1,G}(B_{R})\ \mbox{with}\ v\geqslant\psi_{1}\ a.e.\ \mbox{in}\ \
B_{R}\,,\\\\[1.42271pt] &w_{1}&\geqslant\psi_{1},w_{1}\geqslant 0\ \ \ \ \ \
a.e.\ \mbox{in}\ \ B_{R}\,,\\\\[1.42271pt] &w_{1}&=u\ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \mbox{on}\ \ \partial B_{R}\,,\end{array}\right.$

$\left\\{\begin{array}[]{r@{\ \ }c@{\ \ }ll}&\int_{B_{R}}&a(x,Dw_{2})\cdot
D(v-w_{2})dx\geqslant 0,\,,\\\\[1.42271pt] &\mbox{for}&\ \ \forall\ v\in
w_{2}+W_{0}^{1,G}(B_{R})\ \mbox{with}\ v\geqslant\psi_{1}\ a.e.\ \mbox{in}\ \
B_{R}\,,\\\\[1.42271pt] &w_{2}&\geqslant\psi_{1},w_{2}\geqslant 0\ \ \ \ \ \
a.e.\ \mbox{in}\ \ B_{R}\,,\\\\[1.42271pt] &w_{2}&=w_{1}\ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \mbox{on}\ \ \partial B_{R}\,,\end{array}\right.$

$\left\\{\begin{array}[]{r@{\ \ }c@{\ \
}ll}&\int_{B_{\frac{R}{2}}}&\overline{a}_{B_{\frac{R}{2}}}(Dw_{3})\cdot
D(v-w_{3})dx\geqslant 0,\ \,,\\\\[1.42271pt] &\mbox{for}&\ \ \forall\ v\in
w_{3}+W_{0}^{1,G}(B_{\frac{R}{2}})\ \mbox{with}\ v\geqslant\psi_{1}\ a.e.\
\mbox{in}\ \ B_{\frac{R}{2}}\,,\\\\[1.42271pt] &w_{3}&\geqslant\psi_{1}\ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ a.e.\ \mbox{in}\ \ B_{\frac{R}{2}}\,,\\\\[1.42271pt]
&w_{3}&=w_{2}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{on}\ \ \partial
B_{\frac{R}{2}}\,,\end{array}\right.$

$\left\\{\begin{array}[]{r@{\ \ }c@{\ \
}ll}-\operatorname{div}\left(\overline{a}_{B_{\frac{R}{2}}}(Dw_{4})\right)&=&-\operatorname{div}\left(\overline{a}_{B_{\frac{R}{2}}}(D\psi_{1})\right)\
\ \ \ \ \ \mbox{in}\ \ B_{\frac{R}{2}}\,,\\\\[1.42271pt] w_{4}&=&w_{3}\ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{on}\ \ \partial
B_{\frac{R}{2}}\,,\\\\[1.42271pt] \end{array}\right.$

$\left\\{\begin{array}[]{r@{\ \ }c@{\ \
}ll}-\operatorname{div}\left(\overline{a}_{B_{\frac{R}{2}}}(Dw_{5})\right)&=&0\
\ \ \ \ \mbox{in}\ \ B_{\frac{R}{2}}\,,\\\\[1.42271pt] w_{5}&=&w_{4}\ \ \
\mbox{on}\ \ \partial B_{\frac{R}{2}}\,.\\\\[1.42271pt] \end{array}\right.$

In order to transfer the double obstacle problems to the single one, we make
use of Corollary 3.2 to get the comparison estimate

$\fint_{B_{R}}|Du-Dw_{1}|\operatorname{d}\\!x\leqslant
c\left[\frac{|\mu|(\overline{B_{R}})}{R^{n-1}}+R\fint_{B_{R}}|\operatorname{div}a(x,D\psi_{2})|dx\right]^{\frac{1}{i_{g}}}.$
(3.13)

Using Lemma 3.3 to remove the inhomogeneity,

$\fint_{B_{R}}|Dw_{1}-Dw_{2}|\operatorname{d}\\!x\leqslant
c\left[R\fint_{B_{R}}|\operatorname{div}a(x,D\psi_{2})|dx\right]^{\frac{1}{i_{g}}}.$
(3.14)

Then we use Lemma 3.4 to obtain

$\displaystyle\fint_{B_{\frac{R}{2}}}|Dw_{2}-Dw_{3}|dx$
$\displaystyle\leqslant$ $\displaystyle
c\omega(R)^{\frac{1}{1+s_{g}}}\left\\{\fint_{B_{R}}|Dw_{2}|dx+G^{-1}\left[\fint_{B_{R}}[G(|D\psi_{1}|)+G(|\psi_{1}|)]dx\right]\right\\}$
(3.15) $\displaystyle\leqslant$ $\displaystyle
c\omega(R)^{\frac{1}{1+s_{g}}}\left\\{\fint_{B_{R}}|Du|dx+\left[\frac{|\mu|(\overline{B_{R}})}{R^{n-1}}+R\fint_{B_{R}}|\operatorname{div}a(x,D\psi_{2})|dx\right]^{\frac{1}{i_{g}}}\right\\}$
$\displaystyle+$ $\displaystyle
c\omega(R)^{\frac{1}{1+s_{g}}}\left\\{G^{-1}\left[\fint_{B_{R}}[G(|D\psi_{1}|)+G(|\psi_{1}|)]dx\right]\right\\}.$

In the subsequent step, to effectively transition to an obstacle-free problem,
we make use of Lemma 3.5 to to get

$\fint_{B_{\frac{R}{2}}}|Dw_{3}-Dw_{4}|dx\leqslant
c\left(R\fint_{B_{\frac{R}{2}}}(|\operatorname{div}\left(\overline{a}_{B_{\frac{R}{2}}}(D\psi_{1})\right)|+1)dx\right)^{\frac{1}{i_{g}}}.$
(3.16)

We now proceed to simplify the problem by reducing it to a homogeneous
equation. An application of Lemma 3.6 with
$f=-\operatorname{div}\left(\overline{a}_{B_{\frac{R}{2}}}(D\psi_{1})\right)$
and $g=0$ implies that

$\fint_{B_{\frac{R}{2}}}|Dw_{4}-Dw_{5}|dx\leqslant
c\left(R\fint_{B_{\frac{R}{2}}}(|\operatorname{div}\left(\overline{a}_{B_{\frac{R}{2}}}(D\psi_{1})\right)|+1)dx\right)^{\frac{1}{i_{g}}}.$
(3.17)

Applying Lemma 3.7, there exists $\beta\in(0,1)$ such that

$\displaystyle\fint_{B_{\rho}}|Dw_{5}-(Dw_{5})_{B_{\rho}}|dx$
$\displaystyle\leqslant$ $\displaystyle
c\left(\frac{\rho}{R}\right)^{\beta}\fint_{B_{\frac{R}{2}}}|Dw_{5}-(Dw_{5})_{B_{\frac{R}{2}}}|dx$
(3.18) $\displaystyle\leqslant$ $\displaystyle
c\left(\frac{\rho}{R}\right)^{\beta}\fint_{B_{\frac{R}{2}}}|Dw_{5}-Du|+|Du-(Du)_{B_{R}}|dx.$

Finally combining (3.13)-(3.18) yields

$\displaystyle\fint_{B_{\rho}}|Du-(Du)_{B_{\rho}}|dx\leqslant\fint_{B_{\rho}}|Du-(Dw_{5})_{B_{\rho}}|dx$
$\displaystyle\leqslant$ $\displaystyle\fint_{B_{\rho}}(|Du-
Dw_{1}|+|Dw_{1}-Dw_{2}|+|Dw_{2}-Dw_{3}|$ $\displaystyle+$
$\displaystyle|Dw_{3}-Dw_{4}|+|Dw_{4}-Dw_{5}|+|Dw_{5}-(Dw_{5})_{B_{\rho}}|)dx$
$\displaystyle\leqslant$ $\displaystyle
c\left(\frac{R}{\rho}\right)^{n}\left[\left[\frac{|\mu|(\overline{B_{\frac{R}{2}})}}{R^{n-1}}+R\fint_{B_{R}}|\operatorname{div}a(x,D\psi_{2})|dx\right]^{\frac{1}{i_{g}}}+\left(R\fint_{B_{\frac{R}{2}}}(|\operatorname{div}\left(\overline{a}_{B_{\frac{R}{2}}}(D\psi_{1})\right)|+1)dx\right)^{\frac{1}{i_{g}}}\right]$
$\displaystyle+$ $\displaystyle
c\left(\frac{R}{\rho}\right)^{n}\omega(R)^{\frac{1}{1+s_{g}}}\left\\{\fint_{B_{R}}|Du|dx+\left[\frac{|\mu|(\overline{B_{R}})}{R^{n-1}}+R\fint_{B_{R}}|\operatorname{div}a(x,D\psi_{2})|dx\right]^{\frac{1}{i_{g}}}\right\\}$
$\displaystyle+$ $\displaystyle
c\left(\frac{R}{\rho}\right)^{n}\omega(R)^{\frac{1}{1+s_{g}}}G^{-1}\left[\fint_{B_{R}}[G(|D\psi_{1}|)+G(|\psi_{1}|)]dx\right]$
$\displaystyle+$ $\displaystyle
c\left(\frac{\rho}{R}\right)^{\beta}\left[\fint_{B_{\frac{R}{2}}}(|Dw_{5}-Du|+|Du-(Du)_{B_{R}}|)dx\right]$
$\displaystyle\leqslant$ $\displaystyle
c\left(\frac{\rho}{R}\right)^{\beta}\fint_{B_{R}}|Du-(Du)_{B_{R}}|dx$
$\displaystyle+$ $\displaystyle
c\left(\frac{R}{\rho}\right)^{n}\left[\left[\frac{|\mu|(\overline{B_{R}})}{R^{n-1}}+R\fint_{B_{R}}|\operatorname{div}a(x,D\psi_{2})|dx\right]^{\frac{1}{i_{g}}}+\left(R\fint_{B_{R}}\left(\frac{g(|D\psi_{1}|)}{|D\psi_{1}|}|D^{2}\psi_{1}|+1\right)dx\right)^{\frac{1}{i_{g}}}\right]$
$\displaystyle+$ $\displaystyle
c\left(\frac{R}{\rho}\right)^{n}\omega(R)^{\frac{1}{1+s_{g}}}\left\\{\fint_{B_{R}}|Du|dx+G^{-1}\left[\fint_{B_{R}}[G(|D\psi_{1}|)+G(|\psi_{1}|)]dx\right]\right\\},$

where we used the fact $\omega\leqslant 2L$ and the condition (1.4) in the
last step. ∎

##  4 The proof of gradient estimates

The focus of this section lies in deriving pointwise and oscillation estimates
for the gradients of solutions utilizing the sharp maximal function estimates.
We initiate the analysis by establishing a pointwise estimate of the
fractional maximal operator through the utilization of precise iteration
methods.

###### Proof of Theorem 1.7.

Proof of (1.13) We define

$B_{i}:=B\left(x,\frac{R}{H^{i}}\right)=B(x,R_{i}),\ \ \mbox{for}\ \ \
i=0,1,2,...,\ \ $ $A_{i}:=\fint_{B_{i}}|Du-(Du)_{B_{i}}|dx,\ \
k_{i}:=|(Du)_{B_{i}}-S|,\ \ \ S\in\mathbb{R}^{n}.$

We take $H=H(data)>1$ large enough to have

$c\left(\frac{1}{H}\right)^{\beta}\leq\frac{1}{4}$

with $\beta$ as in Lemma 3.7 and using Lemma 3.8 gives

$\displaystyle\fint_{B_{i+1}}|Du-(Du)_{B_{i+1}}|\operatorname{d}\\!\xi$
$\displaystyle\leqslant$
$\displaystyle\frac{1}{4}\fint_{B_{i}}|Du-(Du)_{B_{i}}|\operatorname{d}\\!\xi+cH^{n}\left\\{\left[\frac{|\mu|(\overline{B_{i}})}{R_{i}^{n-1}}\right]^{\frac{1}{i_{g}}}+\left[\frac{D\Psi_{1}(B_{i})}{R_{i}^{n-1}}\right]^{\frac{1}{i_{g}}}+\left[\frac{D\Psi_{2}(B_{i})}{R_{i}^{n-1}}\right]^{\frac{1}{i_{g}}}\right\\}$
$\displaystyle+$ $\displaystyle
cH^{n}\omega(R_{i})^{\frac{1}{1+s_{g}}}\left\\{\fint_{B_{i}}|Du|d\xi+G^{-1}\left[\fint_{B_{i}}[G(|D\psi_{1}|)+G(|\psi_{1}|)]d\xi\right]\right\\}.$

Our next step involves reducing the value of $R_{0}$-in a way depending only
on $data$ and $\omega(\cdot)$\- to get

$cH^{n}\omega(R_{i})^{\frac{1}{1+s_{g}}}\leqslant
cH^{n}\omega(R_{0})^{\frac{1}{1+s_{g}}}\leqslant\frac{1}{4},$

which together with the following estimate

$\fint_{B_{i}}|Du|d\xi\leqslant\fint_{B_{i}}|Du-(Du)_{B_{i}}|d\xi+k_{i}+|S|,$

we derive that

$\displaystyle A_{i+1}$ $\displaystyle\leqslant$
$\displaystyle\frac{1}{2}A_{i}+c\left\\{\left[\frac{|\mu|(\overline{B_{i}})}{R_{i}^{n-1}}\right]^{\frac{1}{i_{g}}}+\left[\frac{D\Psi_{1}(B_{i})}{R_{i}^{n-1}}\right]^{\frac{1}{i_{g}}}+\left[\frac{D\Psi_{2}(B_{i})}{R_{i}^{n-1}}\right]^{\frac{1}{i_{g}}}\right\\}$
(4.1) $\displaystyle+$ $\displaystyle
c[\omega(R_{i})]^{\frac{1}{1+s_{g}}}\left\\{k_{i}+|S|+G^{-1}\left[\fint_{B_{i}}[G(|D\psi_{1}|)+G(|\psi_{1}|)]d\xi\right]\right\\}$

whenever $i\geqslant 0$. On the other hand, we calculate

$\displaystyle|k_{i+1}-k_{i}|$ $\displaystyle\leq$
$\displaystyle|(Du)_{B_{i+1}}-(Du)_{B_{i}}|$ $\displaystyle\leq$
$\displaystyle\fint_{B_{i+1}}|Du-(Du)_{B_{i}}|\operatorname{d}\\!\xi$
$\displaystyle\leq$ $\displaystyle
H^{n}\fint_{B_{i}}|Du-(Du)_{B_{i}}|\operatorname{d}\\!\xi=H^{n}A_{i},$

from which we see that for $m\in\mathbb{N}$,

$k_{m+1}=\sum_{i=0}^{m}(k_{i+1}-k_{i})+k_{0}\leq
H^{n}\sum_{i=0}^{m}A_{i}+k_{0}.$ (4.2)

In this step, we perform a summation of (4.1) across the range of indices
$i\in\left\\{0,...,m-1\right\\}$, leading us to derive the subsequent
inequality

$\displaystyle\sum_{i=1}^{m}A_{i}$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\sum_{i=0}^{m-1}A_{i}+c\sum_{i=0}^{m-1}\left\\{\left[\frac{|\mu|(\overline{B_{i}})}{R_{i}^{n-1}}\right]^{\frac{1}{i_{g}}}+\left[\frac{D\Psi_{1}(B_{i})}{R_{i}^{n-1}}\right]^{\frac{1}{i_{g}}}+\left[\frac{D\Psi_{2}(B_{i})}{R_{i}^{n-1}}\right]^{\frac{1}{i_{g}}}\right\\}$
$\displaystyle+$ $\displaystyle
c\sum_{i=0}^{m-1}[\omega(R_{i})]^{\frac{1}{1+s_{g}}}\left\\{k_{i}+|S|+G^{-1}\left[\fint_{B_{i}}[G(|D\psi_{1}|)+G(|\psi_{1}|)]d\xi\right]\right\\}.$

Consequently,

$\displaystyle\sum_{i=1}^{m}A_{i}$ $\displaystyle\leq$ $\displaystyle
A_{0}+2c\sum_{i=0}^{m-1}\left\\{\left[\frac{|\mu|(\overline{B_{i}})}{R_{i}^{n-1}}\right]^{\frac{1}{i_{g}}}+\left[\frac{D\Psi_{1}(B_{i})}{R_{i}^{n-1}}\right]^{\frac{1}{i_{g}}}+\left[\frac{D\Psi_{2}(B_{i})}{R_{i}^{n-1}}\right]^{\frac{1}{i_{g}}}\right\\}$
$\displaystyle+$ $\displaystyle
c\sum_{i=0}^{m-1}[\omega(R_{i})]^{\frac{1}{1+s_{g}}}\left\\{k_{i}+|S|+G^{-1}\left[\fint_{B_{i}}[G(|D\psi_{1}|)+G(|\psi_{1}|)]d\xi\right]\right\\}.$

For every integer $m\geqslant 1$ we employ (4.2) to gain

$\displaystyle k_{m+1}$ $\displaystyle\leq$ $\displaystyle
cA_{0}+ck_{0}+c\sum_{i=0}^{m-1}\left\\{\left[\frac{|\mu|(\overline{B_{i}})}{R_{i}^{n-1}}\right]^{\frac{1}{i_{g}}}+\left[\frac{D\Psi_{1}(B_{i})}{R_{i}^{n-1}}\right]^{\frac{1}{i_{g}}}+\left[\frac{D\Psi_{2}(B_{i})}{R_{i}^{n-1}}\right]^{\frac{1}{i_{g}}}\right\\}$
(4.3) $\displaystyle+$ $\displaystyle
c\sum_{i=0}^{m-1}[\omega(R_{i})]^{\frac{1}{1+s_{g}}}\left\\{k_{i}+|S|+G^{-1}\left[\fint_{B_{i}}[G(|D\psi_{1}|)+G(|\psi_{1}|)]d\xi\right]\right\\}.$

Taking into consideration the explicit definition of $A_{0}$, we obtain

$\displaystyle k_{m+1}$ (4.4) $\displaystyle\leq$ $\displaystyle
c\fint_{B_{R}}|Du-(Du)_{B_{R}}|+|Du-S|\operatorname{d}\\!\xi$ $\displaystyle+$
$\displaystyle
c\sum_{i=0}^{m-1}\left\\{\left[\frac{|\mu|(\overline{B_{i}})}{R_{i}^{n-1}}\right]^{\frac{1}{i_{g}}}+\left[\frac{D\Psi_{1}(B_{i})}{R_{i}^{n-1}}\right]^{\frac{1}{i_{g}}}+\left[\frac{D\Psi_{2}(B_{i})}{R_{i}^{n-1}}\right]^{\frac{1}{i_{g}}}\right\\}$
(4.5) $\displaystyle+$ $\displaystyle
c\sum_{i=0}^{m-1}[\omega(R_{i})]^{\frac{1}{1+s_{g}}}\left\\{k_{i}+|S|+G^{-1}\left[\fint_{B_{i}}[G(|D\psi_{1}|)+G(|\psi_{1}|)]d\xi\right]\right\\}$
(4.6)

for every $m\geqslant 0.$ By choosing $S=0$ in the inequality mentioned
earlier and subsequently multiplying both sides by $R_{m+1}^{1-\alpha}$ ,
while considering the given restrictions $\alpha\in[0,1]$ and $R_{m+1}\leq
R_{i}$ for $0\leqslant i\leqslant m+1$, we obtain

$\displaystyle R_{m+1}^{1-\alpha}k_{m+1}$ $\displaystyle\leq$ $\displaystyle
cR^{1-\alpha}\fint_{B_{R}}|Du|\operatorname{d}\\!\xi+c\sum_{i=0}^{m-1}R_{i}^{1-\alpha}\left\\{\left[\frac{|\mu|(\overline{B_{i}})}{R_{i}^{n-1}}\right]^{\frac{1}{i_{g}}}+\left[\frac{D\Psi_{1}(B_{i})}{R_{i}^{n-1}}\right]^{\frac{1}{i_{g}}}+\left[\frac{D\Psi_{2}(B_{i})}{R_{i}^{n-1}}\right]^{\frac{1}{i_{g}}}\right\\}$
$\displaystyle+$ $\displaystyle
c\sum_{i=0}^{m-1}R_{i}^{1-\alpha}[\omega(R_{i})]^{\frac{1}{1+s_{g}}}\left\\{k_{i}+G^{-1}\left[\fint_{B_{i}}[G(|D\psi_{1}|)+G(|\psi_{1}|)]d\xi\right]\right\\}$

and therefore

$\displaystyle R_{m+1}^{1-\alpha}k_{m+1}$ (4.7) $\displaystyle\leq$
$\displaystyle
c\sum_{i=0}^{m}\left\\{\left[\frac{|\mu|(\overline{B_{i}})}{R_{i}^{n-1-i_{g}(1-\alpha)}}\right]^{\frac{1}{i_{g}}}+\left[\frac{D\Psi_{1}(B_{i})}{R_{i}^{n-1-i_{g}(1-\alpha)}}\right]^{\frac{1}{i_{g}}}+\left[\frac{D\Psi_{2}(B_{i})}{R_{i}^{n-1-i_{g}(1-\alpha)}}\right]^{\frac{1}{i_{g}}}\right\\}$
$\displaystyle+$ $\displaystyle
cR^{1-\alpha}\fint_{B_{R}}|Du|\operatorname{d}\\!\xi+c\sum_{i=0}^{m}R_{i}^{1-\alpha}[\omega(R_{i})]^{\frac{1}{1+s_{g}}}\left\\{k_{i}+G^{-1}\left[\fint_{B_{i}}[G(|D\psi_{1}|)+G(|\psi_{1}|)]d\xi\right]\right\\}.$
(4.8)

By employing the definition of Wolff potential, we proceed to estimate the
final term on the right-hand side in equation (4.7), yielding

$\displaystyle\sum_{i=0}^{\infty}\left[\frac{|\mu|(\overline{B_{i}})}{R_{i}^{n-1-i_{g}(1-\alpha)}}\right]^{\frac{1}{i_{g}}}$
(4.9) $\displaystyle\leq$ $\displaystyle\frac{c}{\log
2}\int_{R}^{2R}\left[\frac{|\mu|(\overline{B_{\rho}})}{\rho^{n-1-i_{g}(1-\alpha)}}\right]^{\frac{1}{i_{g}}}\frac{\operatorname{d}\\!\rho}{\rho}+\sum_{i=0}^{\infty}\frac{c}{\log
H}\int_{R_{i+1}}^{R_{i}}\left[\frac{|\mu|(\overline{B_{\rho}})}{\rho^{n-1-i_{g}(1-\alpha)}}\right]^{\frac{1}{i_{g}}}\frac{\operatorname{d}\\!\rho}{\rho}$
$\displaystyle\leq$ $\displaystyle
cW_{1-\alpha+\frac{\alpha}{i_{g}+1},i_{g}+1}^{\mu}(x,2R).$

At the last step, we utilize the assumption $|\mu|(B_{2R})<+\infty$ to deduce
that the condition $|\mu|(\partial B_{\rho})>0$ can hold for only a countable
number of radii $\rho\in(R,2R)$. Likewise, we have

$\sum_{i=0}^{\infty}\left[\frac{D\Psi_{1}(B_{i})}{R_{i}^{n-1-i_{g}(1-\alpha)}}\right]^{\frac{1}{i_{g}}}\leqslant
cW_{1-\alpha+\frac{\alpha}{i_{g}+1},i_{g}+1}^{[\psi_{1}]}(x,2R),$
$\sum_{i=0}^{\infty}\left[\frac{D\Psi_{2}(B_{i})}{R_{i}^{n-1-i_{g}(1-\alpha)}}\right]^{\frac{1}{i_{g}}}\leqslant
cW_{1-\alpha+\frac{\alpha}{i_{g}+1},i_{g}+1}^{[\psi_{2}]}(x,2R),$
$\sum_{i=0}^{\infty}\left[\omega(R_{i})\right]^{\frac{1}{1+s_{g}}}\leqslant
c\int_{0}^{2R}[\omega(\rho)]^{\frac{1}{1+s_{g}}}\frac{d\rho}{\rho},$
$\displaystyle\sum_{i=0}^{\infty}R_{i}^{1-\alpha}\left[\omega(R_{i})\right]^{\frac{1}{1+s_{g}}}G^{-1}\left[\fint_{B_{i}}[G(|D\psi|)+G(|\psi|)]d\xi\right]$
$\displaystyle\leqslant
c\int_{0}^{2R}[\omega(\rho)]^{\frac{1}{1+s_{g}}}G^{-1}\left[\fint_{B_{\rho}}[G(|D\psi_{1}|)+G(|\psi_{1}|)]d\xi\right]\frac{d\rho}{\rho^{\alpha}}.$

Consequently,

$R_{m+1}^{1-\alpha}k_{m+1}\leqslant
cM+c\sum_{i=0}^{m}R_{i}^{1-\alpha}k_{i}\left[\omega(R_{i})\right]^{\frac{1}{1+s_{g}}},$
(4.10)

where

$\displaystyle M:$ $\displaystyle=$ $\displaystyle
c\left[W_{1-\alpha+\frac{\alpha}{i_{g}+1},i_{g}+1}^{\mu}(x,2R)+W_{1-\alpha+\frac{\alpha}{i_{g}+1},i_{g}+1}^{[\psi_{1}]}(x,2R)+W_{1-\alpha+\frac{\alpha}{i_{g}+1},i_{g}+1}^{[\psi_{2}]}(x,2R)\right]$
$\displaystyle+$ $\displaystyle
cR^{1-\alpha}\fint_{B_{R}}|Du|d\xi+c\int_{0}^{2R}[\omega(\rho)]^{\frac{1}{1+s_{g}}}G^{-1}\left[\fint_{B_{\rho}}[G(|D\psi_{1}|)+G(|\psi_{1}|)]d\xi\right]\frac{d\rho}{\rho^{\alpha}}.$

Using the same arguments used in [39], we prove by induction that

$R_{m+1}^{1-\alpha}k_{m+1}\leqslant(c+c^{*})M$ (4.11)

holds for every $m\geqslant 0$, some positive constants $c,c^{*}>1$.

Now we define

$C_{m}:=R_{m}^{1-\alpha}A_{m}=R_{m}^{1-\alpha}\fint_{B_{m}}|Du-(Du)_{B_{m}}|d\xi$
$h_{m}:=\fint_{B_{m}}|Du|d\xi.$

Then it’s easy to obtain

$\displaystyle R_{m}^{1-\alpha}h_{m}$ $\displaystyle=$ $\displaystyle
R_{m}^{1-\alpha}\fint_{B_{m}}|Du|d\xi$ $\displaystyle\leq$ $\displaystyle
R_{m}^{1-\alpha}\fint_{B_{m}}|Du-(Du)_{B_{m}}|+|(Du)_{B_{m}}|d\xi$
$\displaystyle=$ $\displaystyle R_{m}^{1-\alpha}k_{m}+C_{m}$
$\displaystyle\leq$ $\displaystyle CM+C_{m}$

with $M$ as in (4.10). Consequently, our focus narrows down to finding an
upper bound for $C_{m}$. By employing (4.9) and recalling the definition of
$M$ in (4.10), we obtain

$\displaystyle\left[\frac{|\mu|(\overline{B_{i}})}{R_{i}^{n-1}}\right]^{\frac{1}{i_{g}}}$
$\displaystyle\leq$ $\displaystyle
CR_{i}^{\alpha-1}W_{1-\alpha+\frac{\alpha}{i_{g}+1},i_{g}+1}^{\mu}(x,2R)$
$\displaystyle\leq$ $\displaystyle CR_{i}^{\alpha-1}M.$

Likewise, we have

$\left[\frac{D\Psi_{1}(B_{i})}{R_{i}^{n-1}}\right]^{\frac{1}{i_{g}}}\leqslant
cR_{i}^{\alpha-1}M,$
$\left[\frac{D\Psi_{2}(B_{i})}{R_{i}^{n-1}}\right]^{\frac{1}{i_{g}}}\leqslant
cR_{i}^{\alpha-1}M,$
$\left[\omega(R_{i})\right]^{\frac{1}{1+s_{g}}}G^{-1}\left[\fint_{B_{i}}[G(|D\psi_{1}|)+G(|\psi_{1}|)]d\xi\right]\leqslant
cR_{i}^{\alpha-1}M.$

Consequently, with reference to (4.1), we establish

$A_{m+1}\leq\frac{1}{2}A_{m}+cR_{m}^{\alpha-1}M+ck_{m}.$

Subsequently, we employ $\eqref{mguji}$ to obtain

$A_{m+1}\leq\frac{1}{2}A_{m}+cR_{m}^{\alpha-1}M.$

then multiply both sides by $R_{m+1}^{1-\alpha}$, we get

$\displaystyle C_{m+1}$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\left(\frac{R_{m+1}}{R_{m}}\right)^{1-\alpha}C_{m}+C\left(\frac{R_{m+1}}{R_{m}}\right)^{1-\alpha}M$
(4.12) $\displaystyle\leq$ $\displaystyle\frac{1}{2}C_{m}+C_{1}M.$

Then applying the same arguments used in [39], we prove by induction that

$C_{m}\leq 2C_{1}M$ (4.13)

holds whenever $m\geqslant 0$. Let us now consider the case where $r\leqslant
R$ and select the integer $i\geqslant 0$ satisfying $R_{i+1}\leqslant
r\leqslant R_{i}$, then we have

$\displaystyle r^{1-\alpha}\fint_{B_{r}}|Du|\operatorname{d}\\!\xi$
$\displaystyle\leq$
$\displaystyle\left(\frac{R_{i}}{R_{i+1}}\right)^{n}R_{i}^{1-\alpha}\fint_{B_{i}}|Du|\operatorname{d}\\!\xi$
$\displaystyle\leq$ $\displaystyle CH^{n}R_{i}^{1-\alpha}h_{i}$
$\displaystyle\leq$ $\displaystyle CM.$

With the definition of $M$ and the restricted maximal operator in mind, we
obtain in turn

$\displaystyle M_{1-\alpha,R}(|Du|)(x)$ $\displaystyle\leqslant$
$\displaystyle
c\left[W_{1-\alpha+\frac{\alpha}{i_{g}+1},i_{g}+1}^{\mu}(x,2R)+W_{1-\alpha+\frac{\alpha}{i_{g}+1},i_{g}+1}^{[\psi_{1}]}(x,2R)+W_{1-\alpha+\frac{\alpha}{i_{g}+1},i_{g}+1}^{[\psi_{2}]}(x,2R)\right]$
$\displaystyle+$ $\displaystyle
cR^{1-\alpha}\fint_{B_{R}}|Du|d\xi+c\int_{0}^{2R}[\omega(\rho)]^{\frac{1}{1+s_{g}}}G^{-1}\left[\fint_{B_{\rho}}[G(|D\psi_{1}|)+G(|\psi_{1}|)]d\xi\right]\frac{d\rho}{\rho^{\alpha}}.$

Moreover, because of

$M_{\alpha,R}^{\\#}(u)(x)\leqslant M_{1-\alpha,R}(|Du|)(x),$

then we have

$\displaystyle M_{\alpha,R}^{\\#}(u)(x)+M_{1-\alpha,R}(Du)(x)$
$\displaystyle\leqslant$ $\displaystyle
c\left[W_{1-\alpha+\frac{\alpha}{i_{g}+1},i_{g}+1}^{\mu}(x,2R)+W_{1-\alpha+\frac{\alpha}{i_{g}+1},i_{g}+1}^{[\psi_{1}]}(x,2R)+W_{1-\alpha+\frac{\alpha}{i_{g}+1},i_{g}+1}^{[\psi_{2}]}(x,2R)\right]$
$\displaystyle+$ $\displaystyle
cR^{1-\alpha}\fint_{B_{R}}|Du|d\xi+c\int_{0}^{2R}[\omega(\rho)]^{\frac{1}{1+s_{g}}}G^{-1}\left[\fint_{B_{\rho}}[G(|D\psi_{1}|)+G(|\psi_{1}|)]d\xi\right]\frac{d\rho}{\rho^{\alpha}}.$

Proof of (1.15) We define

$\widehat{A}_{i}:=R_{i}^{-\alpha}\fint_{B_{i}}|Du-(Du)_{B_{i}}|\operatorname{d}\\!\xi$
(4.14)

Using Lemma 3.8 (multiply both sides by $R_{i+1}^{-\alpha}$) we obtain

$\displaystyle\widehat{A}_{i+1}$ $\displaystyle\leqslant$ $\displaystyle
c\left(\frac{R_{i+1}}{R_{i}}\right)^{\beta-\alpha}\widehat{A}_{i}+c\left(\frac{R_{i}}{R_{i+1}}\right)^{n+\alpha}\left\\{\left[\frac{|\mu|(\overline{B_{i}})}{R_{i}^{n-1+\alpha
i_{g}}}\right]^{\frac{1}{i_{g}}}+\left[\frac{D\Psi_{1}(B_{i})}{R_{i}^{n-1+\alpha
i_{g}}}\right]^{\frac{1}{i_{g}}}+\left[\frac{D\Psi_{2}(B_{i})}{R_{i}^{n-1+\alpha
i_{g}}}\right]^{\frac{1}{i_{g}}}\right\\}$ $\displaystyle+$ $\displaystyle
c\left(\frac{R_{i}}{R_{i+1}}\right)^{n+\alpha}\frac{1}{R_{i}^{\alpha}}\omega(R_{i})^{\frac{1}{1+s_{g}}}\left\\{\fint_{B_{i}}|Du|d\xi+G^{-1}\left[\fint_{B_{i}}[G(|D\psi_{1}|)+G(|\psi_{1}|)]d\xi\right]\right\\}.$

We choose a value of $H=H(n,i_{a},s_{a},\widehat{\alpha},\beta)>1$ that is
sufficiently large to obtain

$c\left(\frac{R_{i+1}}{R_{i}}\right)^{\beta-\alpha}=c\left(\frac{1}{H}\right)^{\beta-\alpha}\leqslant
c\left(\frac{1}{H}\right)^{\beta-\widehat{\alpha}}\leqslant\frac{1}{2}$

and

$\frac{|\mu|(\overline{B_{i}})}{R_{i}^{n-1+\alpha i_{g}}}\leqslant
H^{n-1+\alpha i_{g}}\frac{|\mu|(B_{i-1})}{R_{i-1}^{n-1+\alpha i_{g}}}.$

Due to assumption (1.14), we have

$\frac{[\omega(R_{i})]^{\frac{1}{1+s_{g}}}}{R_{i}^{\alpha}}\leqslant\frac{[\omega(R_{i})]^{\frac{1}{1+s_{g}}}}{R_{i}^{\widehat{\alpha}}}\leqslant
c_{0},$

then according to the definition of restricted maximal operator, we derive
that

$\displaystyle\widehat{A}_{i+1}$ $\displaystyle\leqslant$
$\displaystyle\frac{1}{2}\widehat{A}_{i}+c\left\\{\left[M_{1-\alpha
i_{g},R}(\mu)(x)\right]^{\frac{1}{i_{g}}}+\left[\overline{M}_{1-\alpha
i_{g},R}(\psi_{1})(x)\right]^{\frac{1}{i_{g}}}+\left[\overline{M}_{1-\alpha
i_{g},R}(\psi_{2})(x)\right]^{\frac{1}{i_{g}}}+\fint_{B_{i}}|Du|d\xi\right\\}$
$\displaystyle+$ $\displaystyle
c\frac{1}{R_{i}^{\alpha}}\omega(R_{i})^{\frac{1}{1+s_{g}}}\left\\{G^{-1}\left[\fint_{B_{i}}[G(|D\psi_{1}|)+G(|\psi_{1}|)]d\xi\right]\right\\}$

for every $i\geqslant 0$. Moreover, from the case $\alpha=1$ of inequality
(1.13), we have

$\displaystyle\fint_{B_{i}}|Du|d\xi$ (4.15) $\displaystyle\leqslant$
$\displaystyle
c\left[\fint_{B_{R}}|Du|d\xi+W_{\frac{1}{i_{g}+1},i_{g}+1}^{\mu}(x,2R)+W_{\frac{1}{i_{g}+1},i_{g}+1}^{[\psi_{1}]}(x,2R)+W_{\frac{1}{i_{g}+1},i_{g}+1}^{[\psi_{2}]}(x,2R)\right]$
$\displaystyle+$ $\displaystyle
c\int_{0}^{2R}[\omega(\rho)]^{\frac{1}{1+s_{g}}}G^{-1}\left[\fint_{B_{\rho}}[G(|D\psi_{1}|)+G(|\psi_{1}|)]d\xi\right]\frac{d\rho}{\rho}$
$\displaystyle:=$ $\displaystyle cM^{*}.$

Therefore, by combining with the two previous inequalities, we obtain

$\displaystyle\widehat{A}_{i+1}$ $\displaystyle\leqslant$
$\displaystyle\frac{1}{2}\widehat{A}_{i}+c\left\\{\left[M_{1-\alpha
i_{g},R}(\mu)(x)\right]^{\frac{1}{i_{g}}}+\left[\overline{M}_{1-\alpha
i_{g},R}(\psi_{1})(x)\right]^{\frac{1}{i_{g}}}+\left[\overline{M}_{1-\alpha
i_{g},R}(\psi_{2})(x)\right]^{\frac{1}{i_{g}}}+M^{*}\right\\}$
$\displaystyle+$ $\displaystyle
c\frac{1}{R_{i}^{\alpha}}\omega(R_{i})^{\frac{1}{1+s_{g}}}\left\\{G^{-1}\left[\fint_{B_{i}}[G(|D\psi_{1}|)+G(|\psi_{1}|)]d\xi\right]\right\\}.$

Iterating the previous relation, we conclude

$\displaystyle\widehat{A}_{i}\leqslant 2^{-i}\widehat{A}_{0}$ $\displaystyle+$
$\displaystyle c\sum_{j=0}^{i-1}2^{-j}\left\\{\left[M_{1-\alpha
i_{g},R}(\mu)(x)\right]^{\frac{1}{i_{g}}}+\left[\overline{M}_{1-\alpha
i_{g},R}(\psi_{1})(x)\right]^{\frac{1}{i_{g}}}+\left[\overline{M}_{1-\alpha
i_{g},R}(\psi_{2})(x)\right]^{\frac{1}{i_{g}}}+M^{*}\right\\}$
$\displaystyle+$ $\displaystyle
c\sum_{j=0}^{i-1}2^{-j}\frac{1}{R_{j}^{\alpha}}\omega(R_{j})^{\frac{1}{1+s_{g}}}\left\\{G^{-1}\left[\fint_{B_{j}}[G(|D\psi_{1}|)+G(|\psi_{1}|)]d\xi\right]\right\\}$

holds for every $i\geqslant 1$. Subsequently, analogous to (4.9), we derive

$\displaystyle\sum_{j=0}^{i-1}2^{-j}\frac{1}{R_{j}^{\alpha}}\omega(R_{j})^{\frac{1}{1+s_{g}}}\left\\{G^{-1}\left[\fint_{B_{j}}[G(|D\psi_{1}|)+G(|\psi_{1}|)]d\xi\right]\right\\}$
$\displaystyle\leqslant$ $\displaystyle
c\int_{0}^{2R}[\omega(\rho)]^{\frac{1}{1+s_{g}}}G^{-1}\left[\fint_{B_{\rho}}[G(|D\psi_{1}|)+G(|\psi_{1}|)]d\xi\right]\frac{d\rho}{\rho^{1+\alpha}}.$

By recalling (4.14) and (2.1), it is straightforward to deduce that

$\displaystyle\sup_{i\geqslant 0}\widehat{A}_{i}$ $\displaystyle\leqslant$
$\displaystyle
c\left\\{R^{-\alpha}\fint_{B_{R}}|Du|\operatorname{d}\\!\xi+\left[M_{1-\alpha
i_{g},R}(\mu)(x)\right]^{\frac{1}{i_{g}}}+\left[\overline{M}_{1-\alpha
i_{g},R}(\psi_{1})(x)\right]^{\frac{1}{i_{g}}}+\left[\overline{M}_{1-\alpha
i_{g},R}(\psi_{2})(x)\right]^{\frac{1}{i_{g}}}\right\\}$ $\displaystyle+$
$\displaystyle
c\left\\{M^{*}+\int_{0}^{2R}[\omega(\rho)]^{\frac{1}{1+s_{g}}}G^{-1}\left[\fint_{B_{\rho}}[G(|D\psi_{1}|)+G(|\psi_{1}|)]d\xi\right]\frac{d\rho}{\rho^{1+\alpha}}\right\\}.$

For every $\rho\in(0,R]$, let $i\in\mathbb{N}$ be such that
$R_{i+1}<\rho\leqslant R_{i}$, then we gain

$\displaystyle\rho^{-\alpha}\fint_{B_{\rho}}|Du-(Du)_{B_{\rho}}|\operatorname{d}\\!\xi$
$\displaystyle\leqslant$ $\displaystyle
c\frac{R_{i}^{n}}{\rho^{n}}R_{i+1}^{-\alpha}\fint_{B_{i}}|Du-(Du)_{B_{i}}|\operatorname{d}\\!\xi$
$\displaystyle\leqslant$ $\displaystyle c\sup_{i\geqslant 0}\widehat{A}_{i},$

from which we derive that

$\displaystyle M_{\alpha,R}^{\\#}(Du)(x)$ $\displaystyle\leqslant$
$\displaystyle
c\left\\{R^{-\alpha}\fint_{B_{R}}|Du|\operatorname{d}\\!\xi+\left[M_{1-\alpha
i_{g},R}(\mu)(x)\right]^{\frac{1}{i_{g}}}+\left[\overline{M}_{1-\alpha
i_{g},R}(\psi_{1})(x)\right]^{\frac{1}{i_{g}}}+\left[\overline{M}_{1-\alpha
i_{g},R}(\psi_{2})(x)\right]^{\frac{1}{i_{g}}}\right\\}$ $\displaystyle+$
$\displaystyle
c\left\\{W_{\frac{1}{i_{g}+1},i_{g}+1}^{\mu}(x,2R)+W_{\frac{1}{i_{g}+1},i_{g}+1}^{[\psi_{1}]}(x,2R)+W_{\frac{1}{i_{g}+1},i_{g}+1}^{[\psi_{2}]}(x,2R)\right\\}$
$\displaystyle+$ $\displaystyle
c\int_{0}^{2R}[\omega(\rho)]^{\frac{1}{1+s_{g}}}G^{-1}\left[\fint_{B_{\rho}}[G(|D\psi_{1}|)+G(|\psi_{1}|)]d\xi\right]\frac{d\rho}{\rho^{1+\alpha}}.$

This completes the proof of Theorem 1.7. ∎

###### Proof of Theorem 1.8.

At first we give the proof of the estimate (1.17).

We choose $S=0$ and make use of (4.4), we conclude

$\displaystyle k_{m+1}$ $\displaystyle\leq$ $\displaystyle
c\fint_{B_{R}(x_{0})}|Du-(Du)_{B_{R}(x_{0})}|+|Du|\operatorname{d}x$
$\displaystyle+$ $\displaystyle
c\sum_{i=0}^{m-1}\left\\{\left[\frac{|\mu|(\overline{B_{R_{i}}}(x_{0}))}{R_{i}^{n-1}}\right]^{\frac{1}{i_{g}}}+\left[\frac{D\Psi_{1}(B_{R_{i}}(x_{0}))}{R_{i}^{n-1}}\right]^{\frac{1}{i_{g}}}+\left[\frac{D\Psi_{2}(B_{R_{i}}(x_{0}))}{R_{i}^{n-1}}\right]^{\frac{1}{i_{g}}}\right\\}$
$\displaystyle+$ $\displaystyle
c\sum_{i=0}^{m-1}[\omega(B_{i})]^{\frac{1}{1+s_{g}}}\left\\{\fint_{B_{R_{i}}(x_{0})}|Du|dx+G^{-1}\left[\fint_{B_{R_{i}}(x_{0})}[G(|D\psi_{1}|)+G(|\psi_{1}|)]dx\right]\right\\}.$

On the other hand, we observe

$\sum_{i=0}^{+\infty}\left[\frac{|\mu|(\overline{B_{R_{i}}}(x_{0}))}{R_{i}^{n-1}}\right]^{\frac{1}{i_{g}}}\leqslant
cW_{\frac{1}{i_{g}+1},i_{g}+1}^{\mu}(x_{0},2R),$
$\sum_{i=0}^{+\infty}\left[\frac{D\Psi_{1}(B_{R_{i}}(x_{0}))}{R_{i}^{n-1}}\right]^{\frac{1}{i_{g}}}\leqslant
cW_{\frac{1}{i_{g}+1},i_{g}+1}^{[\psi_{1}]}(x_{0},2R),$
$\sum_{i=0}^{+\infty}\left[\frac{D\Psi_{2}(B_{R_{i}}(x_{0}))}{R_{i}^{n-1}}\right]^{\frac{1}{i_{g}}}\leqslant
cW_{\frac{1}{i_{g}+1},i_{g}+1}^{[\psi_{2}]}(x_{0},2R),$
$\sum_{i=0}^{+\infty}\left[\omega(R_{i})\right]^{\frac{1}{1+s_{g}}}\leqslant
c\int_{0}^{2R}[\omega(\rho)]^{\frac{1}{1+s_{g}}}\frac{d\rho}{\rho}\leqslant
c,$
$\displaystyle\sum_{i=0}^{+\infty}\left[\omega(R_{i})\right]^{\frac{1}{1+s_{g}}}G^{-1}\left[\fint_{B_{R_{i}}(x_{0})}[G(|D\psi_{1}|)+G(|\psi_{1}|)]dx\right]$
$\displaystyle\leqslant
c\int_{0}^{2R}[\omega(\rho)]^{\frac{1}{1+s_{g}}}G^{-1}\left[\fint_{B_{\rho}(x_{0})}[G(|D\psi_{1}|)+G(|\psi_{1}|)]dx\right]\frac{d\rho}{\rho}.$

Combining (4.15) with above estimates, it follows that

$\displaystyle|Du(x_{0})|=\lim_{m\rightarrow\infty}k_{m+1}$
$\displaystyle\leqslant$ $\displaystyle
c\left[\fint_{B_{R}(x_{0})}|Du|dx+W_{\frac{1}{i_{g}+1},i_{g}+1}^{\mu}(x_{0},2R)+W_{\frac{1}{i_{g}+1},i_{g}+1}^{[\psi_{1}]}(x_{0},2R)+W_{\frac{1}{i_{g}+1},i_{g}+1}^{[\psi_{2}]}(x_{0},2R)\right]$
$\displaystyle+$ $\displaystyle
c\int_{0}^{2R}[\omega(\rho)]^{\frac{1}{1+s_{g}}}G^{-1}\left[\fint_{B_{\rho}(x_{0})}[G(|D\psi_{1}|)+G(|\psi_{1}|)]dx\right]\frac{d\rho}{\rho}.$

Next we prove the estimate (1.18). For every $x,y\in B_{\frac{R}{4}}(x_{0})$,
we define

$r_{i}:=\frac{r}{H^{i}},\ \ \ \ r\leqslant\frac{R}{2},\ \ \ \
k_{i}=|(Du)_{B_{r_{i}}(x)}-S|,\ \ \ \
\overline{k_{i}}=|(Du)_{B_{r_{i}}(y)}-S|.$

Taking use of (4.4) again, we have

$\displaystyle k_{m+1}$ $\displaystyle\leqslant$ $\displaystyle
c\fint_{B_{r}(x)}(|Du-(Du)_{B_{r}(x)}|+|Du-S|)\operatorname{d}\\!\xi$
$\displaystyle+$ $\displaystyle
cr^{\alpha}\sum_{i=0}^{m-1}\left\\{\left[\frac{|\mu|(\overline{B_{r_{i}}}(x))}{r_{i}^{n-1+\alpha
i_{g}}}\right]^{\frac{1}{i_{g}}}+\left[\frac{D\Psi_{1}(B_{r_{i}}(x))}{r_{i}^{n-1+\alpha
i_{g}}}\right]^{\frac{1}{i_{g}}}+\left[\frac{D\Psi_{2}(B_{r_{i}}(x))}{r_{i}^{n-1+\alpha
i_{g}}}\right]^{\frac{1}{i_{g}}}\right\\}$ $\displaystyle+$ $\displaystyle
cr^{\alpha}\sum_{i=0}^{m-1}\frac{1}{r_{i}^{\alpha}}[\omega(r_{i})]^{\frac{1}{1+s_{g}}}\left\\{\fint_{B_{r_{i}}(x)}|Du|d\xi+G^{-1}\left[\fint_{B_{r_{i}}(x)}[G(|D\psi_{1}|)+G(|\psi_{1}|)]d\xi\right]\right\\}.$

Moreover,

$\sum_{i=0}^{+\infty}\left[\frac{|\mu|(\overline{B_{r_{i}}(x)})}{r_{i}^{n-1+\alpha
i_{g}}}\right]^{\frac{1}{i_{g}}}\leqslant cW_{\frac{1-\alpha
i_{g}}{1+i_{g}},i_{g}+1}^{\mu}(x,2r),$
$\sum_{i=0}^{+\infty}\left[\frac{D\Psi_{1}(B_{r_{i}}(x))}{r_{i}^{n-1+\alpha
i_{g}}}\right]^{\frac{1}{i_{g}}}\leqslant cW_{\frac{1-\alpha
i_{g}}{1+i_{g}},i_{g}+1}^{[\psi_{1}]}(x,2r),$
$\sum_{i=0}^{+\infty}\left[\frac{D\Psi_{2}(B_{r_{i}}(x))}{r_{i}^{n-1+\alpha
i_{g}}}\right]^{\frac{1}{i_{g}}}\leqslant cW_{\frac{1-\alpha
i_{g}}{1+i_{g}},i_{g}+1}^{[\psi_{2}]}(x,2r),$
$\sum_{i=0}^{+\infty}\frac{1}{r_{i}^{\alpha}}\left[\omega(r_{i})\right]^{\frac{1}{1+s_{g}}}\leqslant
c\int_{0}^{2r}[\omega(\rho)]^{\frac{1}{1+s_{g}}}\frac{d\rho}{\rho^{1+\alpha}}\leqslant
c,$
$\displaystyle\sum_{i=0}^{+\infty}\frac{1}{r_{i}^{\alpha}}\left[\omega(R_{i})\right]^{\frac{1}{1+s_{g}}}G^{-1}\left[\fint_{B_{r_{i}}(x_{0})}[G(|D\psi_{1}|)+G(|\psi_{1}|)]d\xi\right]$
$\displaystyle\leqslant
c\int_{0}^{2r}[\omega(\rho)]^{\frac{1}{1+s_{g}}}G^{-1}\left[\fint_{B_{\rho}(x)}[G(|D\psi_{1}|)+G(|\psi_{1}|)]d\xi\right]\frac{d\rho}{\rho^{1+\alpha}}.$

Combining (4.15) with the previous estimates, we obtain

$\displaystyle k_{m+1}$ $\displaystyle\leqslant$ $\displaystyle
c\fint_{B_{r}(x)}(|Du-(Du)_{B_{r}(x)}|+|Du-S|)\operatorname{d}\\!\xi$
$\displaystyle+$ $\displaystyle
cr^{\alpha}\left[W^{\mu}_{-\alpha+\frac{1+\alpha}{1+i_{g}},i_{g}+1}(x,R)+W^{[\psi_{1}]}_{-\alpha+\frac{1+\alpha}{1+i_{g}},i_{g}+1}(x,R)+W^{[\psi_{2}]}_{-\alpha+\frac{1+\alpha}{1+i_{g}},i_{g}+1}(x,R)\right]$
$\displaystyle+$ $\displaystyle
cr^{\alpha}\left\\{\fint_{B_{r}(x)}|Du|d\xi+\int_{0}^{R}[\omega(\rho)]^{\frac{1}{1+s_{g}}}G^{-1}\left[\fint_{B_{\rho}(x)}[G(|D\psi_{1}|)+G(|\psi_{1}|)]d\xi\right]\frac{d\rho}{\rho^{1+\alpha}}\right\\}.$

If $x$ is a Lebesgue’s point of $Du$, then let $m\rightarrow\infty$, we derive

$\displaystyle|Du(x)-S|$ $\displaystyle=$
$\displaystyle\lim_{m\rightarrow\infty}k_{m+1}$ $\displaystyle\leqslant$
$\displaystyle
c\fint_{B_{r}(x)}(|Du-(Du)_{B_{r}(x)}|+|Du-S|)\operatorname{d}\\!\xi$
$\displaystyle+$ $\displaystyle
cr^{\alpha}\left[W^{\mu}_{-\alpha+\frac{1+\alpha}{1+i_{g}},i_{g}+1}(x,R)+W^{[\psi_{1}]}_{-\alpha+\frac{1+\alpha}{1+i_{g}},i_{g}+1}(x,R)+W^{[\psi_{2}]}_{-\alpha+\frac{1+\alpha}{1+i_{g}},i_{g}+1}(x,R)\right]$
$\displaystyle+$ $\displaystyle
cr^{\alpha}\left\\{\fint_{B_{r}(x)}|Du|d\xi+\int_{0}^{R}[\omega(\rho)]^{\frac{1}{1+s_{g}}}G^{-1}\left[\fint_{B_{\rho}(x)}[G(|D\psi_{1}|)+G(|\psi_{1}|)]d\xi\right]\frac{d\rho}{\rho^{1+\alpha}}\right\\}.$

If $y$ is a Lebesgue’s point of $Du$, a parallel outcome ensues. Subsequently,
coupling with the two previous estimates enables us to deduce

$\displaystyle|Du(x)-Du(y)|$ $\displaystyle\leqslant$ $\displaystyle
c\fint_{B_{r}(x)}(|Du-(Du)_{B_{r}(x)}|+|Du-S|)\operatorname{d}\\!\xi$
$\displaystyle+$ $\displaystyle
cr^{\alpha}\left[W^{\mu}_{-\alpha+\frac{1+\alpha}{1+i_{g}},i_{g}+1}(x,R)+W^{[\psi_{1}]}_{-\alpha+\frac{1+\alpha}{1+i_{g}},i_{g}+1}(x,R)+W^{[\psi_{2}]}_{-\alpha+\frac{1+\alpha}{1+i_{g}},i_{g}+1}(x,R)\right]$
$\displaystyle+$ $\displaystyle
cr^{\alpha}\left\\{\fint_{B_{r}(x)}|Du|d\xi+\int_{0}^{R}[\omega(\rho)]^{\frac{1}{1+s_{g}}}G^{-1}\left[\fint_{B_{\rho}(x)}[G(|D\psi_{1}|)+G(|\psi_{1}|)]d\xi\right]\frac{d\rho}{\rho^{1+\alpha}}\right\\}$
$\displaystyle+$ $\displaystyle
c\fint_{B_{r}(y)}(|Du-(Du)_{B_{r}(y)}|+|Du-S|)\operatorname{d}\\!\xi$
$\displaystyle+$ $\displaystyle
cr^{\alpha}\left[W^{\mu}_{-\alpha+\frac{1+\alpha}{1+i_{g}},i_{g}+1}(y,R)+W^{[\psi_{1}]}_{-\alpha+\frac{1+\alpha}{1+i_{g}},i_{g}+1}(y,R)+W^{[\psi_{2}]}_{-\alpha+\frac{1+\alpha}{1+i_{g}},i_{g}+1}(y,R)\right]$
$\displaystyle+$ $\displaystyle
cr^{\alpha}\left\\{\fint_{B_{r}(y)}|Du|d\xi+\int_{0}^{R}[\omega(\rho)]^{\frac{1}{1+s_{g}}}G^{-1}\left[\fint_{B_{\rho}(y)}[G(|D\psi_{1}|)+G(|\psi_{1}|)]d\xi\right]\frac{d\rho}{\rho^{1+\alpha}}\right\\}.$

We now choose

$S:=(Du)_{B_{3r}(x)},\ \ \ r:=\dfrac{|x-y|}{2},$

it’s easy to see that $B_{r}(y)\subseteq B_{3r}(x)$ and therefore

$\displaystyle\fint_{B_{r}(x)}(|Du-(Du)_{B_{r}(x)}|+|Du-S|)\operatorname{d}\\!\xi+\fint_{B_{r}(y)}(|Du-(Du)_{B_{r}(y)}|+|Du-S|)\operatorname{d}\\!\xi$
$\displaystyle\leqslant$ $\displaystyle
c(n)\fint_{B_{3r}(x)}|Du-(Du)_{B_{3r}(x)}|\operatorname{d}\\!\xi.$

Now notice that $x,y\in B_{\frac{R}{4}}(x_{0})$, so
$|x-y|\leqslant\frac{R}{2}$ and then $B_{3r}(x)\subseteq
B_{\frac{3R}{4}}(x)\subseteq B_{R}(x_{0})$. Therefore apply (1.15) to obtain

$\displaystyle\fint_{B_{3r}(x)}|Du-(Du)_{B_{3r}(x)}|\operatorname{d}\\!\xi$
$\displaystyle\leqslant$ $\displaystyle
cr^{\alpha}M^{\\#}_{\alpha,\frac{3R}{4}}(Du)(x)$ $\displaystyle\leqslant$
$\displaystyle
c\left(\frac{r}{R}\right)^{\alpha}\fint_{B_{\frac{3R}{4}}(x)}|Du|\operatorname{d}\\!\xi$
$\displaystyle+$ $\displaystyle cr^{\alpha}\left\\{\left[M_{1-\alpha
i_{g},\frac{3R}{4}}(\mu)(x)\right]^{\frac{1}{i_{g}}}+\left[\overline{M}_{1-\alpha
i_{g},\frac{3R}{4}}(\psi_{1})(x)\right]^{\frac{1}{i_{g}}}+\left[\overline{M}_{1-\alpha
i_{g},\frac{3R}{4}}(\psi_{2})(x)\right]^{\frac{1}{i_{g}}}\right\\}$
$\displaystyle+$ $\displaystyle
cr^{\alpha}\left\\{W_{\frac{1}{i_{g}+1},i_{g}+1}^{\mu}(x,2R)+W_{\frac{1}{i_{g}+1},i_{g}+1}^{[\psi_{1}]}(x,2R)+W_{\frac{1}{i_{g}+1},i_{g}+1}^{[\psi_{2}]}(x,2R)\right\\}$
$\displaystyle+$ $\displaystyle
cr^{\alpha}\int_{0}^{2R}[\omega(\rho)]^{\frac{1}{1+s_{g}}}G^{-1}\left[\fint_{B_{\rho}(x)}[G(|D\psi_{1}|)+G(|\psi_{1}|)]d\xi\right]\frac{d\rho}{\rho^{1+\alpha}}.$

Moreover, because of (4.15), we have

$\displaystyle\fint_{B_{r}(x)}|Du|d\xi$ $\displaystyle\leqslant$
$\displaystyle
c\left[\fint_{B_{\frac{3R}{4}}(x)}|Du|d\xi+W_{\frac{1}{i_{g}+1},i_{g}+1}^{\mu}(x,\frac{3R}{2})+W_{\frac{1}{i_{g}+1},i_{g}+1}^{[\psi_{1}]}(x,\frac{3R}{2})+W_{\frac{1}{i_{g}+1},i_{g}+1}^{[\psi_{2}]}(x,\frac{3R}{2})\right]$
$\displaystyle+$ $\displaystyle
c\int_{0}^{\frac{3R}{2}}[\omega(\rho)]^{\frac{1}{1+s_{g}}}G^{-1}\left[\fint_{B_{\rho}(x)}[G(|D\psi_{1}|)+G(|\psi_{1}|)]d\xi\right]\frac{d\rho}{\rho},$
$\displaystyle\fint_{B_{r}(y)}|Du|d\xi$ $\displaystyle\leqslant$
$\displaystyle
c\left[\fint_{B_{\frac{3R}{4}}(y)}|Du|d\xi+W_{\frac{1}{i_{g}+1},i_{g}+1}^{\mu}(y,\frac{3R}{2})+W_{\frac{1}{i_{g}+1},i_{g}+1}^{[\psi_{1}]}(y,\frac{3R}{2})+W_{\frac{1}{i_{g}+1},i_{g}+1}^{[\psi_{2}]}(y,\frac{3R}{2})\right]$
$\displaystyle+$ $\displaystyle
c\int_{0}^{\frac{3R}{2}}[\omega(\rho)]^{\frac{1}{1+s_{g}}}G^{-1}\left[\fint_{B_{\rho}(y)}[G(|D\psi_{1}|)+G(|\psi_{1}|)]d\xi\right]\frac{d\rho}{\rho}.$

Due to the preceding estimates, we derive

$\displaystyle|Du(x)-Du(y)|\leqslant
c\left(\frac{r}{R}\right)^{\alpha}\fint_{B_{R}(x_{0})}|Du|\operatorname{d}\\!\xi$
$\displaystyle+$ $\displaystyle cr^{\alpha}\left\\{\left[M_{1-\alpha
i_{g},\frac{3R}{4}}(\mu)(x)\right]^{\frac{1}{i_{g}}}+\left[\overline{M}_{1-\alpha
i_{g},\frac{3R}{4}}(\psi_{1})(x)\right]^{\frac{1}{i_{g}}}+\left[\overline{M}_{1-\alpha
i_{g},\frac{3R}{4}}(\psi_{2})(x)\right]^{\frac{1}{i_{g}}}\right\\}$
$\displaystyle+$ $\displaystyle
cr^{\alpha}\left[W^{\mu}_{-\alpha+\frac{1+\alpha}{1+i_{g}},i_{g}+1}(x,2R)+W^{[\psi_{1}]}_{-\alpha+\frac{1+\alpha}{1+i_{g}},i_{g}+1}(x,2R)+W^{[\psi_{2}]}_{-\alpha+\frac{1+\alpha}{1+i_{g}},i_{g}+1}(x,2R)\right]$
$\displaystyle+$ $\displaystyle
cr^{\alpha}\left[W^{\mu}_{-\alpha+\frac{1+\alpha}{1+i_{g}},i_{g}+1}(y,2R)+W^{[\psi_{1}]}_{-\alpha+\frac{1+\alpha}{1+i_{g}},i_{g}+1}(y,2R)+W^{[\psi_{2}]}_{-\alpha+\frac{1+\alpha}{1+i_{g}},i_{g}+1}(y,2R)\right]$
$\displaystyle+$ $\displaystyle
cr^{\alpha}\left[\int_{0}^{2R}[\omega(\rho)]^{\frac{1}{1+s_{g}}}G^{-1}\left[\fint_{B_{\rho}(x)}[G(|D\psi_{1}|)+G(|\psi_{1}|)]d\xi\right]\frac{d\rho}{\rho^{1+\alpha}}\right]$
$\displaystyle+$ $\displaystyle
cr^{\alpha}\left[\int_{0}^{2R}[\omega(\rho)]^{\frac{1}{1+s_{g}}}G^{-1}\left[\fint_{B_{\rho}(y)}[G(|D\psi_{1}|)+G(|\psi_{1}|)]d\xi\right]\frac{d\rho}{\rho^{1+\alpha}}\right],$

where we used the fact that
$W_{\frac{1}{i_{g}+1},i_{g}+1}^{\mu}(x,2R)\leqslant
cW^{\mu}_{-\alpha+\frac{1+\alpha}{1+i_{g}},i_{g}+1}(x,2R)$ and for other case
there are similar inequalities.

For every $\varepsilon>0$, we know that there exists $0<r\leqslant R$ such
that

$M_{1-\alpha
i_{g},\frac{3R}{4}}(\mu)(x)\leqslant|B_{1}|^{-1}\frac{|\mu|(B_{\frac{3r}{4}}(x))}{\left(\frac{3r}{4}\right)^{n-1+\alpha
i_{g}}}+\varepsilon.$

According to the definition of Wolff potential, we obtain

$\displaystyle\frac{|\mu|(B_{\frac{3r}{4}}(x))}{\left(\frac{3r}{4}\right)^{n-1+\alpha
i_{g}}}$ $\displaystyle=$
$\displaystyle\left[\left(\frac{|\mu|(B_{\frac{3r}{4}}(x))}{\left(\frac{3r}{4}\right)^{n-1+\alpha
i_{g}}}\right)^{\frac{1}{i_{g}}}\frac{1}{-\log(3/4)}\int_{3r/4}^{r}\frac{\operatorname{d}\\!\rho}{\rho}\right]^{i_{g}}$
$\displaystyle\leqslant$ $\displaystyle
C\left[\int_{3r/4}^{r}\left(\frac{|\mu|(B_{\rho}(x))}{{\rho}^{n-1+\alpha
i_{g}}}\right)^{\frac{1}{i_{g}}}\frac{\operatorname{d}\\!\rho}{\rho}\right]^{i_{g}}$
$\displaystyle\leqslant$ $\displaystyle
C\left[W^{\mu}_{-\alpha+\frac{1+\alpha}{1+i_{g}},i_{g}+1}(x,R)\right]^{i_{g}}.$

Likewise,

$\frac{D\Psi_{1}(B_{\frac{3r}{4}}(x))}{\left(\frac{3r}{4}\right)^{n-1+\alpha
i_{g}}}\leqslant
C\left[W^{[\psi_{1}]}_{-\alpha+\frac{1+\alpha}{1+i_{g}},i_{g}+1}(x,R)\right]^{i_{g}},$
$\frac{D\Psi_{2}(B_{\frac{3r}{4}}(x))}{\left(\frac{3r}{4}\right)^{n-1+\alpha
i_{g}}}\leqslant
C\left[W^{[\psi_{2}]}_{-\alpha+\frac{1+\alpha}{1+i_{g}},i_{g}+1}(x,R)\right]^{i_{g}}.$

Finally, we consider the definition of $r$ to derive

$\displaystyle|Du(x)-Du(y)|$ $\displaystyle\leq$ $\displaystyle
c\fint_{B_{R}(x_{0})}|Du|\operatorname{d}\\!\xi\left(\frac{|x-y|}{R}\right)^{\alpha}$
$\displaystyle+$ $\displaystyle
c\left[W^{\mu}_{-\alpha+\frac{1+\alpha}{1+i_{g}},i_{g}+1}(x,2R)+W^{[\psi_{1}]}_{-\alpha+\frac{1+\alpha}{1+i_{g}},i_{g}+1}(x,2R)+W^{[\psi_{2}]}_{-\alpha+\frac{1+\alpha}{1+i_{g}},i_{g}+1}(x,2R)\right]|x-y|^{\alpha}$
$\displaystyle+$ $\displaystyle
c\left[W^{\mu}_{-\alpha+\frac{1+\alpha}{1+i_{g}},i_{g}+1}(y,2R)+W^{[\psi_{1}]}_{-\alpha+\frac{1+\alpha}{1+i_{g}},i_{g}+1}(y,2R)+W^{[\psi_{2}]}_{-\alpha+\frac{1+\alpha}{1+i_{g}},i_{g}+1}(y,2R)\right]|x-y|^{\alpha}$
$\displaystyle+$ $\displaystyle
c\left[\int_{0}^{2R}[\omega(\rho)]^{\frac{1}{1+s_{g}}}G^{-1}\left[\fint_{B_{\rho}(x)}[G(|D\psi_{1}|)+G(|\psi_{1}|)]d\xi\right]\frac{d\rho}{\rho^{1+\alpha}}\right]|x-y|^{\alpha}$
$\displaystyle+$ $\displaystyle
c\left[\int_{0}^{2R}[\omega(\rho)]^{\frac{1}{1+s_{g}}}G^{-1}\left[\fint_{B_{\rho}(y)}[G(|D\psi_{1}|)+G(|\psi_{1}|)]d\xi\right]\frac{d\rho}{\rho^{1+\alpha}}\right]|x-y|^{\alpha}.$

Then we finish the proof of Theorem 1.8. ∎

## Acknowledgments

The authors are supported by the Fundamental Research Funds for the Central
Universities (Grant No. 2682024CX028) and the National Natural Science
Foundation of China (Grant No. 12071229 and 12101452).

## References

  * [1] R.A. Adams, Sobolev Spaces. Academic Press, New York (1975)
  * [2] P. Baroni, Riesz potential estimates for a general class of quasilinear equations, Calc. Var. Partial Differential Equations 53 (3-4) (2015) 803-846.
  * [3] L. Bers, Mathematical Aspects of Subsonic and Transonic Gas Dynamics. Wiley, New York (1958).
  * [4] L. Beck, G. Mingione: Lipschitz bounds and nonuniform ellipticity, Comm. Pure Appl. Math. 73 (2020) 944-1034.
  * [5] Y. Cho, Global gradient estimates for divergence-type elliptic problems involving general nonlinear operators, J. Differ. Equ. 264(2018) 6152-6190.
  * [6] A. Cianchi, V. Maz’ya, Gradient regularity via rearrangements for p-Laplacian type elliptic boundary value problems, J. Eur. Math. Soc. (JEMS) 16 (2014) 571-595.
  * [7] A. Cianchi, V. Maz’ya, Global Lipschitz regularity for a class of quasilinear elliptic equations, Comm. Partial Differential Equations 36 (2011) 100-133.
  * [8] A. Cianchi, V. Maz’ya, Global boundedness of the gradient for a class of nonlinear elliptic systems, Arch. Ration. Mech. Anal. 212 (2014) 129-177.
  * [9] L. Diening, F. Ettwein, Fractional estimates for non-differentiable elliptic systems with general growth, Forum Math, 20(3) (2008) 523-556.
  * [10] F. Duzaar, G. Mingione, Gradient estimates via non-linear potentials, Amer. J. Math. 133 (2011) 1093-1149.
  * [11] F. Duzaar, G. Mingione, Gradient estimates via linear and nonlinear potentials, J. Funct. Anal. 259 (2010) 2961-2998.
  * [12] F. Duzaar, G. Mingione, Gradient continuity estimates, Calc. Var. Partial Differ. Equ. 39 (3-4) (2010) 379-418.
  * [13] E. Giusti, Direct Methods in the Calculus of Variations, World Scientific Publishing Co.,Inc.,River Edge (2003).
  * [14] L. Hedberg, Th.H. Wolff, Thin sets in Nonlinear Potential Theory, Ann. Inst. Fourier (Grenoble) 33 (1983) 161-187.
  * [15] T. Kilpeläinen, J. Malý, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math. 172 (1994) 137-161.
  * [16] T. Kilpeläinen, J. Malý, Degenerate elliptic equations with measure data and nonlinear potentials, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 19 (1992) 591-613.
  * [17] T. Kuusi, G. Mingione, Universal potential estimates, J. Funct. Anal. 262 (2012), 4205-4269.
  * [18] T. Kuusi, G. Mingione, Vectorial nonlinear potential theory, J. Eur. Math. Soc. 20 (2018) 929-1004.
  * [19] T. Kuusi, G. Mingione, Guide to nonlinear potential estimates (English summary), Bull. Math. Sci. 4 (2014) 1-82.
  * [20] T. Kuusi, G. Mingione,Y. Sire, Nonlocal equations with measure data, Comm. Math. Phys. 337 (2015) 1317-1368.
  * [21] T. Kuusi, G. Mingione, Partial regularity and potentials, J.$\acute{E}$c. Polytech. Math. 3 (2016) 309–363.
  * [22] T. Kuusi, G. Mingione, Riesz potentials and nonlinear parabolic equations. Arch. Ration. Mech. Anal. 212 (2014) 727-780.
  * [23] T. Kuusi, G. Mingione, The Wolff gradient bound for degenerate parabolic equations. J. Eur. Math. Soc. (JEMS) 16 (2014) 835-892.
  * [24] T. Kuusi, G. Mingione, Potential estimates and gradient boundedness for nonlinear parabolic systems. Rev. Mat. Iberoam. 28 (2012) 535-576.
  * [25] T. Kuusi, G. Mingione, New perturbation methods for nonlinear parabolic problems. Journal de Math$\acute{e}$matiques Pures et Appliqu$\acute{e}$es (9) 98 (2012) 390-427.
  * [26] G.M. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Ural’tseva for elliptic equations, Comm. Partial Differential Equations 16 (1991) 311-361.
  * [27] G.M. Lieberman, Regularity of solutions to some degenerate double obstacle problems, Indiana Univ. Math. J. 40 (1991) 1009-1028.
  * [28] G. Mingione, Gradient potential estimates, J. Eur. Math. Soc. (JEMS) 13 (2011), 459-486.
  * [29] V.G. Maz’ja, V.P. Havin, A nonlinear potential theory, Uspehi Mat. Nauk 27 (1972) 67-138.
  * [30] L. Ma, Z. Zhang, Wolff type potential estimates for stationary Stokes systems with Dini-BMO coefficients, Commun. Contemp. Math. 23 (2021), no. 7, Paper No. 2050064.
  * [31] M.M. Rao, Z.D. Ren, Theory of Orlicz spaces, Monographs and Textbooks in Pure and Applied Mathematics, 146. Marcel Dekker, Inc., New York, 1991.
  * [32] J.F. Rodrigues, R. Teymurazyan, On the two obstacles problem in Orlicz-Sobolev spaces and applications (English summary), Complex Var. Elliptic Equ. 56 (2011) 769-787.
  * [33] C. Scheven, Gradient potential estimates in non-linear elliptic obstacle problems with measure data, J. Funct. Anal. 262 (2012) 2777-2832.
  * [34] C. Scheven, Elliptic obstacle problems with measure data: potentials and low order regularity, Publ. Mat. 56 (2012) 327-374.
  * [35] N. Trudinger, X. Wang, On the weak continuity of elliptic operators and applications to potential theory, Amer. J. Math. 124 (2002) 369-410.
  * [36] N. Trudinger, X. Wang, Quasilinear elliptic equations with signed measure data, Discrete Contin. Dyn. Syst. 23 (2009) 477-494.
  * [37] Q. Xiong, Z. Zhang, Gradient potential estimates for elliptic obstacle problems, J. Math. Anal. Appl. 495(2021) 124698.
  * [38] Q. Xiong, Z. Zhang, Ma, Lingwei, Riesz potential estimates for problems with Orlicz growth. J. Math. Anal. Appl. 515 (2022), no. 2, Paper No. 126448, 38 pp.
  * [39] Q. Xiong, Z. Zhang, Ma, Lingwei, Gradient potential estimates in elliptic obstacle problems with Orlicz growth. Calc. Var. Partial Differential Equations 61 (2022), no. 3, Paper No. 83, 33 pp.
  * [40] X. Fu, J. Xiao, Q. Xiong, Toward Weighted Lorentz–Sobolev Capacities from Caffarelli–Silvestre Extensions, J. Geom. Anal. 34 (2024), no. 5, Paper No. 124.
  * [41] X. Fu, J. Xiao, Q. Xiong, Dirichlet or Neumann problem for weighted 1-Laplace equation with application to image denoising, J. Geom. Anal. 34 (2024), no. 1, Paper No. 32, 49 pp.
  * [42] F.P. Yao, S. Zhou, Calder$\acute{o}$n-Zygmund estimates for a class of quasilinear elliptic equations, J. Funct. Anal. 272 (2017), 1524-1552.
  * [43] J. Zhang, S. Zheng, Z. Feng, Gradient estimates of general nonlinear singular elliptic equations with measure data, J. Differential Equations 372 (2023), 402-457.