AI 中文总结
该研究证明抛物型Lipschitz函数图像的曲面测度与热测度的绝对连续性等价于抛物型Dini-β条件成立，且这类图像可被正则Lipschitz图像覆盖，为抛物型可求长性提供了定性判定依据。
AI 中文摘要
我们证明：若∂Ω是抛物型Lipschitz函数的图像，则∂Ω的抛物型曲面测度σ关于其热测度绝对连续，当且仅当（平方）Dini-β条件成立。具体而言，（平方）Dini-β条件为：对σ-几乎处处的(X,t)∈∂Ω，有∫₀¹β̂(X,t,r)² (dr/r) < ∞，其中β̂是Jones（L²）β数的抛物型版本。我们还证明，上述条件成立当且仅当该图像可被可数个“正则”Lipschitz图像覆盖，这类图像具有抛物型BMO空间中半阶时间导数形式的额外时间正则性。这支持了如下观点：在抛物型偏微分方程的语境下，被“正则”Lipschitz图像覆盖是定性抛物型可求长性的正确概念。此外，我们还证明：若在热测度零集外满足∫₀¹β̂(X,t,r)² (dr/r) < ∞，则热测度关于曲面测度绝对连续。
英文摘要
We show if $\partial Ω$ is the graph of a parabolic Lipschitz function, then parabolic surface measure $σ$ of $\partial Ω$ is absolutely continuous with respect to its caloric measure if and only if a (square) Dini-$β$ condition is satisfied. More specifically, the (square) Dini-$β$ condition is that \[\int_0^1 \hatβ(X,t,r)^2 \frac{dr}{r} < \infty, \quad \text{$σ$-a.e. } (X,t) \in \partial Ω.\] Here $\hatβ$ is a parabolic version of the Jones ($L^2$) $β$-numbers. We show that these conditions are satisfied if and only if the graph is covered by a countable collection of {\it regular} Lipschitz graphs, that is, graphs with additional in-time regularity in the form of a half order time derivative in the parabolic BMO space. This supports the view that covering by {\it regular} Lipschitz graphs is the right notion for qualitative parabolic rectifiability in the context of parabolic PDEs. We also show that if \[\int_0^1 \hatβ(X,t,r)^2 \frac{dr}{r} < \infty\] up to a set of caloric measure zero then the caloric measure is absolutely continuous with respect to surface measure.
