The Riesz transform and quantitative rectifiability for general Radon measures

© 2017, Springer-Verlag GmbH Germany, part of Springer Nature. In this paper we show that if μ is a Borel measure in Rn+1 with growth of order n, such that the n-dimensional Riesz transform Rμ is bounded in L2(μ) , and B⊂ Rn+1 is a ball with μ(B) ≈ r(B) n such that:(a)there is some n-plane L passing through the center of B such that for some δ> 0 small enough, it holds (Formula presented.)(b)for some constant ε> 0 small enough, (Formula presented.) where mμ,B(Rμ1) stands for the mean of Rμ1 on B with respect to μ, then there exists a uniformly n-rectifiable set Γ , with μ(Γ ∩ B) ≳ μ(B) , and such that μ| Γ is absolutely continuous with respect to Hn| Γ. This result is an essential tool to solve an old question on a two phase problem for harmonic measure in subsequent papers by Azzam, Mourgoglou, Tolsa, and Volberg.