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 $2\le n\le d-2$. 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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