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 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 G is a union of m m open arcs on the boundary of the unit disc D \mathbf {D} , then is ω a , p ( G ) = ω a , p ( G ¯ ) \omega _{a,p}(G)=\omega _{a,p}(\overline {G}) , where ω a , p \omega _{a,p} denotes the p p\mspace {1mu} -harmonic measure? (Strictly speaking he stated this question for the case m = 2 m=2 .) For p = 2 p=2 the positive answer to this question is well known. Recall that for p ≠ 2 p \ne 2 the p 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 1>p>2 . In the proof, using a deep trace result of Jonsson and Wallin, it is first shown that the characteristic function χ G \chi _G is the restriction to ∂ D \partial \mathbf {D} of a Sobolev function from W 1 , p ( C ) W^{1,p}(\mathbf {C}) . For p ≥ 2 p \ge 2 it is no longer true that χ G \chi _G belongs to the trace class. Nevertheless, we are able to show equality for the case m = 1 m=1 of one arc for all 1 > p > ∞ 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.