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

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" $$\underset{\epsilon >0}{\sup }{\int }_E{\left|{\int }_{\{y\in E:|x-y|>\epsilon \}}\frac{x-y}{|x-y{|}^{n+1}}f(y)d{H}^n(y)\right|}^{2}d{H}^n(x)\le C{\int }_E|f{|}^{2}d{H}^n.$$ 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.

References in corpus (2)

• Uniform rectifiability, Calderon-Zygmund operators with odd kernel, and quasiorthogonality
• Boundedness of the square function and rectifiability

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