Failure of almost monotonicity of harmonic measure density ratios at points of vanishing codimension-one density

Technical Summary: Failure of Almost Monotonicity of Harmonic Measure Density Ratios

Problem Statement
This paper investigates the metric and geometric properties of harmonic measure $\omega$ on the boundary $\partial\Omega$ of an arbitrary open set $\Omega \subset \mathbb{R}^{n+1}$. Specifically, it focuses on the behavior of the density ratios $\Theta^n_\omega(x, r) = \omega(B(x, r))/r^n$ at points where the density vanishes. Let $Z$ be the set of points where the limit of these ratios is zero:
$$Z = \left\{ x \in \mathbb{R}^{n+1} : \lim_{r \to 0} \Theta^n_\omega(x, r) = 0 \right\}.$$
While it is known that for $n=1$ (the planar case), the set $Z$ has zero harmonic measure, in higher dimensions ($n \ge 2$), Wolff constructed examples where $\omega(Z) > 0$. The central question addressed is the regularity of the density ratio function $r \mapsto \Theta^n_\omega(x, r)$ on this exceptional set $Z$. The authors ask whether this function exhibits "almost monotonicity" (being almost increasing or almost decreasing) at small scales.

Methodology
The proof proceeds by contradiction. The authors assume that for a subset of $Z$ with positive harmonic measure, the density ratio function is almost monotone. They then construct a scenario where this assumption implies the existence of an $n$-rectifiable set carrying positive harmonic measure within $Z$. This leads to a contradiction, as harmonic measure is known to vanish on $n$-rectifiable subsets of $Z$.

The core technical machinery involves the following steps:

1. Rectifiability Criterion: The proof relies on a variant of a rectifiability criterion (Theorem 3.1), adapted from the second author's previous work [10] and the framework of [1]. This criterion states that if a Radon measure $\mu$ satisfies specific conditions on a ball $B$—namely, polynomial growth, boundedness of the Riesz transform, and small $L^2$ oscillation of the Riesz transform—then a significant portion of the measure is supported on an $n$-rectifiable set.
2. David-Mattila Lattice: To localize the problem and handle the lack of regularity in the domain boundary, the authors utilize the David-Mattila dyadic lattice of cubes associated with the harmonic measure. This allows for a multiscale analysis of the measure's distribution.
3. Modification of the Domain: A critical and delicate part of the argument involves modifying the domain $\Omega$ by removing "bad" cubes from the boundary (using a "good" set of cubes from the lattice). This creates a new domain $\tilde{\Omega}$ where the boundary is sufficiently regular to apply potential theoretic estimates, specifically relating the Green function to the harmonic measure.
4. Touching-Point Argument: To verify the hypothesis of the rectifiability criterion regarding the existence of a smaller ball with a density ratio bounded from below (Hypothesis b in Theorem 3.1), the authors employ a touching-point argument. They identify a point $q$ where the Green function is large and construct a ball $B_1$ tangent to the modified boundary. By comparing the Green functions of the original and modified domains, they establish the necessary lower bound on the density ratio of $B_1$.
5. Connection to Green Function: The estimates for the Riesz transform (Hypotheses a and c) are derived by linking the Riesz transform to the gradient of the Green function, utilizing the almost monotonicity assumption to control the growth of the measure.

Key Contributions and Results

• Theorem 1.1: The main result establishes that for an open set $\Omega \subset \mathbb{R}^{n+1}$ with compact boundary and harmonic measure $\omega$, at $\omega$-almost every point $x \in Z$ (where the density vanishes), the density ratio is not almost monotone. Specifically:
  $$\limsup_{R \to 0} \sup_{0 < r < R} \frac{\Theta^n_\omega(x, r)}{\Theta^n_\omega(x, R)} = \infty.$$
  This implies that the density ratios exhibit arbitrarily large oscillations at sufficiently small scales.

• Corollary 1.1: The result is extended to the set $Z^*$ where only the limit inferior of the density is zero ($\liminf_{r \to 0} \Theta^n_\omega(x, r) = 0$), showing the same oscillatory behavior holds for $\omega$-almost every point in this larger set.

• Technical Tool (Theorem 3.1): The paper provides a refined rectifiability criterion suitable for harmonic measure on general domains, which serves as the primary engine for the contradiction argument.

Significance
The paper claims that this result provides a qualitative converse to the asymptotic regularity enjoyed by harmonic measure on rectifiable pieces of the boundary. On rectifiable sets, the density ratio converges to a positive limit, making the function almost monotone near zero. The failure of almost monotonicity on the set $Z$ highlights the highly irregular, oscillatory nature of harmonic measure at points where the density vanishes.

The authors emphasize that this phenomenon is intrinsic to the singular nature of harmonic measure on such sets and cannot occur if the harmonic measure is mutually absolutely continuous with respect to the $n$-dimensional Hausdorff measure. The work deepens the understanding of the metric properties of harmonic measure in higher dimensions, particularly regarding the structure of exceptional sets where the density vanishes.