Traces of monotone functions in weighted Sobolev spaces

Abstract

Consider monotone functions $u\colon \mathbb{B}^n\to \mathbb{R}$ in the weighted Sobolev space $W^{1,p}(\mathbb{B}^n;w)$, where $n-1<p\leq n$ and $w$ is a weight in the class $A_q$ for some $1\leq q<{{p}/({n-1})}$ which has a certain symmetry property with respect to $\partial \mathbb{B}^n$. We prove that $u$ has nontangential limits at all points of $\partial \mathbb{B}^n$ except possibly those on a set $E$ of weighted $(p,w)$-capacity zero. The proof is based on a new weighted oscillation estimate (Theorem 1) that may be of independent interest. In the special case $w(x)=|1-|x||^\alpha$, the weighted $(p,w)$-capacity of a ball can be easily estimated to conclude that the Hausdorff dimension of the set $E$ is smaller than or equal to $\alpha+n-p$, where $0\leq \alpha<{(p-(n-1))}/{(n-1)}$.