抛物型均匀可修正性由有界热函数的Carleson测度估计刻画
Parabolic uniform rectifiability is characterized by Carleson measure estimates for bounded caloric functions
- University of Alabama（阿拉巴马大学）
- Universitat de Barcelona（巴塞罗那大学）
- University of Missouri（密苏里大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明抛物型均匀可修正性等价于热方程有界解的Carleson测度估计及热测度的日冕分解，在温和假设下建立了新的刻画。
AI中文摘要：
我们证明了一个集合的抛物型均匀可修正性可以由一个内部的PDE性质来刻画，即热方程有界解的Carleson测度估计。特别地，在关于开集$\Omega$及其边界$\partial \Omega$的非常温和的背景假设下，我们证明以下条件是等价的：(i) 量$|\nabla u(\cdot)|^2 \text{dist}(\cdot, \partial\Omega)$是$\Omega$上所有热方程有界解的Carleson测度的密度。(ii) $\Omega$的热测度允许一个日冕分解。(iii) $\partial \Omega$是抛物型均匀可修正的。蕴含关系(iii)推出(i)是引用论文[Bortz, Hoffman, Hofmann, Luna-García, Nyström, Anal. PDE 2023]的主要结果。其余蕴含关系是新的，并且是Garnett、Mourgoglou和Tolsa [Duke Math. J. 2018]在椭圆情形下关于Laplacian的结果的直接抛物型类比；然而，我们的证明需要几个新的想法。特别是，我们必须调整论证以考虑时间滞后，以及（椭圆）均匀可修正性的某些刻画在抛物型情形下要么尚未被发展，要么已被证明不成立的事实。如上所述，我们的背景假设非常温和：我们假设区域满足时间对称容量密度条件并且具有内部螺旋性质，并且边界是（非时间定向的）Ahlfors-David正则的。这些比Bortz、Hofmann、Martell和Nyström在arXiv:2510.22047中的背景假设更弱，并且对于我们的论证所依赖的热函数的边界连续性性质来说本质上是最优的。
英文摘要：
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 $\partial Ω$, we show that the following are equivalent: (i) The quantity $|\nabla u(\cdot)|^2 \text{dist}(\cdot, \partialΩ)$ 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) $\partial Ω$ 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.
