Parabolic uniform rectifiability is characterized by Carleson measure estimates for bounded caloric functions
Abstract
We show that the parabolic uniform rectifiability of a set can be characterized by an interior PDE property, namely Carleson measure estimates for bounded solutions to the heat equation. In particular, under very mild background hypotheses on an open set Ω and its boundary ∂ Ω, we show that the following are equivalent: (i) The quantity |∇ u(·)|2 dist(·, ∂Ω) is the density of a Carleson measure on Ω for all bounded solutions to the heat equation in Ω. (ii) The caloric measure for Ω admits a corona decomposition. (iii) ∂ Ω is parabolic uniformly rectifiable. The implication (iii) implies (i) is the main result of the cited paper [Bortz, Hoffman, Hofmann, Luna-García, Nyström, Anal. PDE 2023]. The remaining implications are new, and are direct parabolic analogues of results of Garnett, Mourgoglou and Tolsa [Duke Math. J. 2018] in the elliptic setting for the Laplacian; however, our proofs require several new ideas. In particular, we must adapt arguments to account both for time-lag, and for the fact that some characterizations of (elliptic) uniform rectifiability have either not been developed or have been shown to be untrue in the parabolic setting. Our background assumptions, as noted above, are very mild: we assume that the domain satisfies the time-symmetric capacity density condition and has interior corkscrews, and that the boundary is (non time-directed) Ahlfors-David regular. These are weaker than the background assumptions of Bortz, Hofmann, Martell and Nyström in arXiv:2510.22047, and are essentially optimal for the boundary continuity properties of caloric functions on which our arguments rely.
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BibTeX
@misc{bortz2026parabolic,
title = {{Parabolic uniform rectifiability is characterized by Carleson measure estimates for bounded caloric functions}},
author = {Simon Bortz and Pablo Hidalgo-Palencia and Steve Hofmann and James Warta},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15374},
eprint = {2609.15374},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15374},
}
RIS
TY - UNPB
TI - Parabolic uniform rectifiability is characterized by Carleson measure estimates for bounded caloric functions
AU - Simon Bortz
AU - Pablo Hidalgo-Palencia
AU - Steve Hofmann
AU - James Warta
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15374
UR - https://arxiv.org/abs/2609.15374
AB - We show that the parabolic uniform rectifiability of a set can be characterized by an interior PDE property, namely Carleson measure estimates for bounded solutions to the heat equation. In particular, under very mild background hypotheses on an open set Ω and its boundary ∂ Ω, we show that the following are equivalent: (i) The quantity |∇ u(·)|^2 dist(·, ∂Ω) is the density of a Carleson measure on Ω for all bounded solutions to the heat equation in Ω. (ii) The caloric measure for Ω admits a corona decomposition. (iii) ∂ Ω is parabolic uniformly rectifiable. The implication (iii) implies (i) is the main result of the cited paper [Bortz, Hoffman, Hofmann, Luna-García, Nyström, Anal. PDE 2023]. The remaining implications are new, and are direct parabolic analogues of results of Garnett, Mourgoglou and Tolsa [Duke Math. J. 2018] in the elliptic setting for the Laplacian; however, our proofs require several new ideas. In particular, we must adapt arguments to account both for time-lag, and for the fact that some characterizations of (elliptic) uniform rectifiability have either not been developed or have been shown to be untrue in the parabolic setting. Our background assumptions, as noted above, are very mild: we assume that the domain satisfies the time-symmetric capacity density condition and has interior corkscrews, and that the boundary is (non time-directed) Ahlfors-David regular. These are weaker than the background assumptions of Bortz, Hofmann, Martell and Nyström in arXiv:2510.22047, and are essentially optimal for the boundary continuity properties of caloric functions on which our arguments rely.
ER -