We show that BMO-solvability implies scale invariant quantitative absolute continuity (specifically, the weak-\(A_{\infty }\) property) of caloric measure with respect to surface measure, for an open set Ω ⊂ ℝn+ 1, assuming as a background hypothesis only that the essential boundary of Ω satisfies an appropriate parabolic version of Ahlfors-David regularity, entailing some backwards in time thickness. Since the weak-\(A_{\infty }\) property of the caloric measure is equivalent to L^p solvability of the initial-Dirichlet problem, we may then deduce that BMO-solvability implies L^p solvability for some finite p.

中文翻译：

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BMO的可测性和绝对测度的连续性

我们表明，BMO-有解意味着尺度不变定量绝对连续性（具体地，weak- \（A _ {\ infty} \）属性）热量测量相对于表面的措施，为一个开集Ω⊂ℝ Ñ + 1，假定作为背景假设，只有Ω的基本边界满足Ahlfors-David正则性的合适的抛物线形式，这需要在时间上向后倾斜。由于热量测量的弱\（A _ {\ infty} \）属性等于初始Dirichlet问题的L ^p可解性，因此我们可以推断出BMO可解性暗示了对于某些有限p的L ^p可解性。