The Riesz transform and quantitative rectifiability for general Radon measures

TL;DR

This paper establishes conditions under which a measure with bounded Riesz transform is supported on a uniformly rectifiable set, advancing understanding of geometric measure theory and harmonic analysis.

Contribution

It provides new criteria linking Riesz transform boundedness and geometric rectifiability for Radon measures in Euclidean space.

Findings

Bounded Riesz transform implies existence of rectifiable subsets

Conditions relate measure's proximity to an n-plane and Riesz transform oscillation

Results facilitate solving problems in harmonic measure theory

Abstract

In this paper we show that if $\mu$ is a Borel measure in $\mathbb R^{n+1}$ with growth of order $n$, so that the $n$-dimensional Riesz transform $R_\mu$ is bounded in $L^2(\mu)$, and $B\subset\mathbb R^{n+1}$ is a ball with $\mu(B)\approx r(B)^n$ such that: (a) there is some $n$-plane $L$ passing through the center of $B$ such that for some $\delta>0$ small enough, it holds $\int_B \frac{dist(x,L)}{r(B)}\,d\mu(x)\leq \delta\,\mu(B),$ (b) for some constant $\epsilon>0$ small enough, $\int_B |R_\mu1(x) - m_{\mu,B}(R_\mu1)|^2\,d\mu(x) \leq \epsilon \,\mu(B)$, where $m_{\mu,B}(R_\mu1)$ stands for the mean of $R_\mu1$ on $B$ with respect to $\mu$; then there exists a uniformly $n$-rectifiable subset $\Gamma$, with $\mu(\Gamma\cap B)\gtrsim \mu(B)$, and so that $\mu|_\Gamma$ is absolutely continuous with respect to $H^n|_\Gamma$. This result is an essential tool to solve an old question on a two phase problem for harmonic measure in a subsequent paper by Azzam, Mourgoglou and Tolsa.