Parabolic Reifenberg Parametrizations and Corona Decompositions, with Applications to Singular Integrals

Abstract

For parabolic Ahlfors regular sets, we show that the bilateral weak geometric lemma (BWGL), bilateral approximation by unions of planes (BAUP), and corona decompositions by $Lip$(1,1/2) graphs (whose defining functions need not have half-order time derivatives in BMO) are equivalent. These geometric conditions do not imply the strong geometric lemma. A key tool is a parabolic version of Reifenberg's theorem for sets with holes, which we prove here. As an application of these geometric results, we prove that a parabolic Ahlfors $(n+2)$-regular measure $μ$ on $\mathbb{R}^{d+1}$, $1\leq n\leq d-1$, is parabolic uniformly rectifiable if its support satisfies any of these equivalent conditions and the parabolic Riesz transform is bounded on $L^2(μ)$.