A problem of Baernstein on the equality of the $p\mspace {1mu}$-harmonic measure of a set and its closure

Abstract:

A. Baernstein II (Comparison of $p\mspace {1mu}$-harmonic measures of subsets of the unit circle, St. Petersburg Math. J. 9 (1998), 543–551, p. 548), posed the following question: If $G$ is a union of $m$ open arcs on the boundary of the unit disc $\mathbf {D}$, then is $\omega _{a,p}(G)=\omega _{a,p}(\overline {G})$, where $\omega _{a,p}$ denotes the $p\mspace {1mu}$-harmonic measure? (Strictly speaking he stated this question for the case $m=2$.) For $p=2$ the positive answer to this question is well known. Recall that for $p \ne 2$ the $p\mspace {1mu}$-harmonic measure, being a nonlinear analogue of the harmonic measure, is not a measure in the usual sense. The purpose of this note is to answer a more general version of Baernstein’s question in the affirmative when $1<p<2$. In the proof, using a deep trace result of Jonsson and Wallin, it is first shown that the characteristic function $\chi _G$ is the restriction to $\partial \mathbf {D}$ of a Sobolev function from $W^{1,p}(\mathbf {C})$. For $p \ge 2$ it is no longer true that $\chi _G$ belongs to the trace class. Nevertheless, we are able to show equality for the case $m=1$ of one arc for all $1<p<\infty$, using a very elementary argument. A similar argument is used to obtain a result for starshaped domains. Finally we show that in a certain sense the equality holds for almost all relatively open sets.