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arXiv · 1904.08407

BMO Solvability and Absolute Continuity of Caloric Measure

Alyssa Genschaw·Steve Hofmann
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Abstract

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 $\Omega \subset \mathbb{R}^{n+1}$, assuming as a background hypothesis only that the essential boundary of $\Omega$ 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 $p$.

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Alyssa Genschaw, Steve Hofmann. 2019-04-16. BMO Solvability and Absolute Continuity of Caloric Measure. https://arxiv.org/abs/1904.08407

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