In this paper, we obtain the following local weighted Lorentz gradient estimates

$$\begin{aligned} g^{-1}\left( {\mathcal {M}}_1(\mu ) \right) \in L_{w,{\text {loc}}}^{q,r}(\Omega ) \Longrightarrow |Du| \in L_{w,{\text {loc}}}^{q,r}(\Omega ) \end{aligned}$$

for the weak solutions of a class of non-homogeneous quasilinear elliptic equations with measure data

$$\begin{aligned} -\text {div} ~\! \left( a\left( \left| \nabla u \right| \right) \nabla u \right) = \mu , \end{aligned}$$

where \(g(t)= t a(t)\) for \(t\ge 0\) and

$$\begin{aligned} {\mathcal {M}}_1(\mu )(x):=\sup _{r>0}\frac{r|\mu |(B_r(x))}{|B_r(x)|}, \quad x\in {\mathbb {R}}^{n}. \end{aligned}$$

Moreover, we remark that two natural and simple examples of functions g(t) in this work are

$$\begin{aligned} g(t)=t^{p-1} ~~(p\text{-Laplace } \text{ equation) }~~~~~~ \text{ and } ~~~~~~ g(t)=t^{p-1 }\log ^\alpha \big ( 1+t\big ) \quad \text{ for }~\alpha > 0. \end{aligned}$$

Actually, the more general and interesting example is related to (p, q)-growth condition by appropriate gluing of the monomials. We remark that our results improve the known results for such equations.

中文翻译：

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一类带有测量数据的拟线性椭圆方程的加权Lorentz梯度估计。

在本文中，我们获得以下局部加权洛伦兹梯度估计

$$ \ begin {aligned} g ^ {-1} \ left（{\ mathcal {M}} _ 1（\ mu）\ right）\ in L_ {w，{\ text {loc}}} ^ {q，r }（\ Omega）\ Longrightarrow | Du | \ in L_ {w，{\ text {loc}}} ^ {q，r}（\ Omega）\ end {aligned} $$

量测数据的一类非齐次拟线性椭圆型方程的弱解

$$ \ begin {aligned}-\ text {div}〜\！\ left（a \ left（\ left | \ nabla u \ right | \ right）\ nabla u \ right）= \ mu，\ end {aligned} $$

其中\（g（t）= ta（t）\）为\（t \ ge 0 \）和

$$ \ begin {aligned} {\ mathcal {M}} _ 1（\ mu）（x）：= \ sup _ {r> 0} \ frac {r | \ mu |（B_r（x））} {| B_r （x）|}，\ quad x \在{\ mathbb {R}} ^ {n}中。\ end {aligned} $$

此外，我们注意到在这项工作中，函数g（t）的两个自然而简单的例子是

$$ \ begin {aligned} g（t）= t ^ {p-1} ~~（p \ text {-Laplace} \ text {公式）} ~~~~~~ \ text {和} ~~~~ ~~ g（t）= t ^ {p-1} \ log ^ \ alpha \ big（1 + t \ big）\ quad \ text {为}〜\ alpha>0。\ end {aligned} $$

实际上，更普遍和有趣的示例是通过适当地胶合单项式元素与（p，  q）-增长条件有关的。我们注意到我们的结果改进了此类方程的已知结果。