Universal potential estimates for $1<p\le 2-\frac{1}{n}$

We extend the so-called universal potential estimates of Kuusi-Mingione type (J. Funct. Anal. 262: 4205–4269, 2012) to the singular case $1<p\le 2-1/n$ for the quasilinear equation with measure data

$$-\mathrm{div}(A(x,\mathrm{\nabla }u))=\mu$$

in a bounded open subset $\mathrm{\Omega }$ of ${\mathbb{R}}^n$, $n\ge 2$, with a finite signed measure $\mu$ in $\mathrm{\Omega }$. The operator $\mathrm{div}(A(x,\mathrm{\nabla }u))$ is modeled after the $p$-Laplacian ${\mathrm{\Delta }}_{p}u:=\mathrm{d}\mathrm{i}\mathrm{v}(|\mathrm{\nabla }u{|}^{p-2}\mathrm{\nabla }u)$, where the nonlinearity $A(x,\xi )$ ( $x,\xi \in {\mathbb{R}}^n$) is assumed to satisfy natural growth and monotonicity conditions of order $p$, as well as certain additional regularity conditions in the $x$-variable.