Uniform Rectifiability and Harmonic Measure III: Riesz transform bounds imply uniform rectifiability of boundaries of 1-sided NTA domains

TL;DR

This paper proves that for certain regular boundaries of domains satisfying specific geometric conditions, a bound on the Riesz transform implies the boundary is uniformly rectifiable, linking harmonic analysis and geometric measure theory.

Contribution

It establishes that Riesz transform bounds on Ahlfors-David regular sets imply uniform rectifiability for boundaries of 1-sided NTA domains, advancing understanding of geometric conditions linked to harmonic analysis.

Findings

Riesz transform bounds imply uniform rectifiability

Boundaries of 1-sided NTA domains are uniformly rectifiable under these conditions

Connects harmonic analysis bounds with geometric regularity

Abstract

Let $E\subset \mathbb{R}^{n+1}$, $n\ge 2$, be a closed, Ahlfors-David regular set of dimension $n$ satisfying the "Riesz Transform bound" $$\sup_{\varepsilon>0}\int_E\left|\int_{\{y\in E:|x-y|>\varepsilon\}}\frac{x-y}{|x-y|^{n+1}} f(y) dH^n(y)\right|^2 dH^n(x) \leq C \int_E|f|^2 dH^n .$$ Assume further that $E$ is the boundary of a domain $\Omega\subset \mathbb{R}^{n+1}$ satisfying the Harnack Chain condition plus an interior (but not exterior) Corkscrew condition. Then $E$ is uniformly rectifiable.