PreprintarXiv licencearXiv
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 ℝd+1, 1≤ n≤ d−1, is parabolic uniformly rectifiable if its support satisfies any of these equivalent conditions and the parabolic Riesz transform is bounded on L2(μ).
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BibTeX
@misc{hoffman2026parabolic,
title = {{Parabolic Reifenberg Parametrizations and Corona Decompositions, with Applications to Singular Integrals}},
author = {John Hoffman and Matthew Hyde and Mihalis Mourgoglou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.42142},
eprint = {2610.42142},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.42142},
}
RIS
TY - UNPB
TI - Parabolic Reifenberg Parametrizations and Corona Decompositions, with Applications to Singular Integrals
AU - John Hoffman
AU - Matthew Hyde
AU - Mihalis Mourgoglou
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.42142
UR - https://arxiv.org/abs/2610.42142
AB - 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 ℝ^d+1, 1≤ n≤ 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).
ER -