Gradient estimates for singular $p$-Laplace type equations with measure data

We are concerned with interior and global gradient estimates for solutions to a class of singular quasilinear elliptic equations with measure data, whose prototype is given by the $p$-Laplace equation $-\Delta_p u=\mu$ with $p\in (1,2)$. The cases when $p\in \big(2-\frac 1 n,2\big)$ and $p\in \big(\frac{3n-2}{2n-1},2-\frac{1}{n}\big]$ were studied in [9] and [22], respectively. In this paper, we improve the results in [22] and address the open case when $p\in \big(1,\frac{3n-2}{2n-1}\big]$. Interior and global modulus of continuity estimates of the gradients of solutions are also established.