The David-Semmes problem in higher codimension: bounded Riesz transforms force uniform rectifiability for 1 < n < d-1

Theorem 1.3: for integers $d\ge4$ and $2\le n\le d-2$, an $n$-AD-regular Radon measure on $\mathbb R^d$ whose hard-truncated Riesz transforms are uniformly bounded $L^2(\mu)\to L^2(\mu;\mathbb R^d)$ is uniformly $n$-rectifiable, with constants depending only on $d,n,C_{AD},C_R$; the conclusion is given as uniform big pieces of Lipschitz images of balls. With the known cases $n=1$ and $n=d-1$ this answers the David-Semmes question in every dimension and codimension. The converse direction and non-integer dimensions are not addressed here.

Theorem 1.3: for integers $d\ge4$ and $2\le n\le d-2$, an $n$-AD-regular Radon measure on $\mathbb R^d$ whose hard-truncated Riesz transforms are uniformly bounded $L^2(\mu)\to L^2(\mu;\mathbb R^d)$ is uniformly $n$-rectifiable, with constants depending only on $d,n,C_{AD},C_R$; the conclusion is given as uniform big pieces of Lipschitz images of balls. With the known cases $n=1$ and $n=d-1$ this answers the David-Semmes question in every dimension and codimension. The converse direction and non-integer dimensions are not addressed here.