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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 L

^{2}

-bounded over all positive hard truncations. The rectifiability bounds depend only on dimension, regularity and operator bounds.

Manuscripts in this family · 1

Riesz transforms and uniform rectifiability in higher codimension

Sep 24, 2026·paper.pdfLean

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.

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