Traces of monotone functions in weighted Sobolev spaces

Consider monotone functions u: B-n --> R in the weighted Sobolev space W-1,W-p (B-n; omega), where n - 1 < p less than or equal to n and omega is a weight in the class A(q) for some 1 < q < p/(n - 1) which has a certain symmetry property with respect to deltaB(n). We prove that u has nontangential limits at all points of deltaB(n) except possibly those on a set E of weighted (p, -)capacity zero. The proof is based on a new weighted oscillation estimate (Theorem 1) that may be of independent interest. In the special case omega(x) = \1 - \x\\(alpha), the weighted (p,omega)-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 < alpha < (p - (n -1))/(n - 1).