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

Abstract

We extend the so-called universal potential estimates of the Kuusi-Mingione type (J.Funct. Anal. 2012) to the singular case $1<p\le 2-1/n$ for the quasilinear equation with measure data \begin{equation*} -\operatorname{div}(A(x,\nabla u))=μ \end{equation*} in a bounded open subset $\Omega$ of ${\mathbb{R}}^n$, $n\ge 2$, with a finite signed measure $\mu$ in $\Omega$. The operator $\mathrm{div}(A(x,\nabla u))$ is modeled after the $p$-Laplacian ${\Delta }_{p}u:=\mathrm{div}(|\nabla u{|}^{p-2}\nabla u)$, 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.