A Counterexample Regarding a Two-phase Problem for Harmonic Measure in VMO

Let Ω+⊂Rn+1 be a vanishing Reifenberg flat domain such that Ω+ and Ω- = Rn+1\Ω+¯ have joint big pieces of chord-arc subdomains and such that the outer unit normal to ∂Ω+ belongs to VMO(ω+), where ω± is the harmonic measure in Ω±. Up to now it was an open question if these conditions imply that logdω-dω+∈VMO(ω+). In this paper we answer this question in the negative by constructing an appropriate counterexample in R2, with the additional property that the outer unit normal to ∂Ω+ is constant ω+-a.e. in ∂Ω+.