Riesz Transform Bounds Imply Parabolic Uniform Rectifiability of Graphs

Abstract

We prove that if a $(k+2)$-Ahlfors–David regular measure $μ$ in parabolic space $\mathbb P^n$ has support equal to a parabolic Lipschitz graph with sufficiently small Lipschitz constant, and the parabolic Riesz transform is bounded on $L^2(μ)$, then $μ$ is parabolic uniformly rectifiable and the graph is regular parabolic Lipschitz. The result holds for every integer $0\leq k\leq n-1$.