Riesz transforms and rectifiability in higher codimension
OpenAI research collection · 1 manuscript · Source checked 2026-10-19
OpenAI's result summary
Resolves the remaining higher-codimension Riesz-transform rectifiability problem: for d ≥ 4 and 2≤ n≤ d-2, an n-Ahlfors–David regular Radon measure on ℝ^(d) is uniformly n-rectifiable whenever its n-dimensional Riesz transform is uniformly L^(2)-bounded over all positive hard truncations. The rectifiability bounds depend only on dimension, regularity and operator bounds.
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MANUSCRIPT 01
Riesz transforms and uniform rectifiability in higher codimension
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≤ n≤ d-2. This gives a positive answer to the David–Semmes…
Full source abstract
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≤ n≤ 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.
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