On an obstacle to the converse of Dahlberg's theorem in high codimensions

TL;DR

This paper explores the relationship between rectifiability and harmonic measure in higher codimension sets, proposing new results and discussing obstacles in extending classical theorems to these complex geometric contexts.

Contribution

It provides initial results supporting the conjecture that harmonic measure characterizes hyperplanes in higher codimension, and analyzes the limitations of current strategies.

Findings

Evidence that $L_\alpha D_\alpha = 0$ only for hyperplanes

Identification of obstacles in extending classical theorems

Discussion of strategies that do not fully resolve the problem

Abstract

It has been recently understood that the harmonic measure on the boundary $E = \partial \Omega$ of a domain $\Omega$ in $\mathbb{R}^n$ is absolutely continuous with respect to the Hausdorff measure $\mathcal{H}^{n - 1}$ on $E$ if and only if the boundary $E$ is rectifiable. Then, by G. David, M. Engelstein, J. Feneuil, S. Mayboroda and other coauthors, a notion of harmonic measure for Ahlfors-regular sets $E$ of higher codimension $n - d$ was developed with the aid of the operator $L_\alpha = -\mbox{div} D_{\alpha}^{-n + d + \alpha} \nabla$, where $\alpha > 0$ and $D_\alpha$ is a certain regularized distance function to the set $E$. A program was launched to establish analogous to the classical case equivalence between rectifiability of the higher-codimensional set $E$ and good relations of the (new) harmonic and Hausdorff measures. The sufficiency of rectifiability for quantitative absolute continuity was only just obtained. For the other direction the main obstacle is to prove that, roughly, the equation $L_\alpha D_\alpha = 0$ is true only when the set $E$ is a hyperplane. In this paper we prove some first results which indicate that the latter conjecture may be true. We also explain that a certain natural strategy to tackle the problem does not work till the end.