Riesz transforms and rectifiability in higher codimension

Resolves the remaining higher-codimension Riesz-transform rectifiability problem: for d ≥ 4 and 2\le n\le d-2, an n-Ahlfors–David regular Radon measure on ℝ^d is uniformly n-rectifiable whenever its n-dimensional Riesz transform is uniformly L²-bounded over all positive hard truncations. The rectifiability bounds depend only on dimension, regularity and operator bounds. (Lean)

Manuscript

Riesz transforms and uniform rectifiability in higher codimension

September 24, 2026 · source · Lean formalization available · cite as \cite{OAI:Riesz-transforms-and-uniform-rectifiability-in-higher-codimension-September-24-2026}

We prove that an n-Ahlfors–David regular Radon measure on ℝ^d is uniformly n-rectifiable if its n-dimensional Riesz transform is uniformly bounded from scalar L^2(\mu) to vector-valued L^2(\mu) over all positive hard truncations, for integers d ≥ 4 and 2\le n\le d-2. This gives a positive answer to the David–Semmes Riesz-transform question in its remaining higher-codimension range. The conclusion gives uniform big pieces of Lipschitz images of Euclidean balls.

Result 081 · Real and complex analysis. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.