The two-phase problem for harmonic measure in VMO via jump formulas for the Riesz transform
Martí Prats, Xavier Tolsa · arXiv (Cornell University) · 2019
Let $\Omega^+\subset\mathbb R^{n+1}$ be an NTA domain and let $\Omega^-= \mathbb R^{n+1}\setminus \overline{\Omega^+}$ be an NTA domain as well. Denote by $\omega^+$ and $\omega^-$ their respective harmonic measures. Assume that $\Omega^+$ is a $\delta$-Reifenberg flat domain, for some $\delta>0$ small enough. In this paper we show that if $\omega^+$ and $\omega^-$ are mutually absolutely continuous and $\log\frac{d\omega^-}{d\omega^+}\in VMO(\omega^+)$, then the inner unit normal of $\Omega^+$ also satisfies $N\in VMO(\omega^+)$. To obtain this result we prove jump formulas for the non-tangential limits of Riesz transforms and other singular integrals which are valid for arbitrary rectifiable sets and have their own interest.