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Summer 2001 Traces of monotone functions in weighted Sobolev spaces
Juan J. Manfredi, Enrique Villamor
Illinois J. Math. 45(2): 403-422 (Summer 2001). DOI: 10.1215/ijm/1258138347


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 $0\leq \alpha<{(p-(n-1))}/{(n-1)}$.


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Juan J. Manfredi. Enrique Villamor. "Traces of monotone functions in weighted Sobolev spaces." Illinois J. Math. 45 (2) 403 - 422, Summer 2001.


Published: Summer 2001
First available in Project Euclid: 13 November 2009

zbMATH: 0988.30015
MathSciNet: MR1878611
Digital Object Identifier: 10.1215/ijm/1258138347

Primary: 31B25
Secondary: 30C65, 35J25, 35J67, 46E35

Rights: Copyright © 2001 University of Illinois at Urbana-Champaign


Vol.45 • No. 2 • Summer 2001
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