Small $A_\infty$ results for Dahlberg-Kenig-Pipher operators in domains with low-dimensional, uniformly rectifiable boundaries


In this talk, I will discuss joint work with M. Engelstein, L. Li, and S. Mayboroda, where we introduce the notion of Dahlberg-Kenig-Pipher operators in the context of domains $\Omega$ with low dimensional boundaries. We show that when the boundary of the domain is uniformly rectifiable with small constant, then elliptic measure $\omega$ associated to this domain is an $A_\infty(d\sigma)$ weight with small constant. As consequence, we show that for $C^1$ domains, $\log (d\omega/d\sigma) \in \mathrm{VMO}$. One of the main difficulties in this context is the lack of outer graphical approximations to $\Omega$, since the exterior of $\Omega$ can be empty.