# BMO Solvability and Absolute Continuity of Caloric Measure

## TL;DR

This paper demonstrates that BMO-solvability of the caloric equation ensures a scale-invariant absolute continuity of caloric measure relative to surface measure, linking boundary regularity to solvability of boundary value problems.

## Contribution

It establishes that BMO-solvability implies the weak-$A_
abla$ property of caloric measure under parabolic Ahlfors-David regularity, leading to $L^p$ solvability.

## Key findings

- BMO-solvability implies weak-$A_
abla$ absolute continuity of caloric measure.
- Weak-$A_
abla$ property is equivalent to $L^p$ solvability of the Dirichlet problem.
- Results hold under parabolic Ahlfors-David regularity of the boundary.

## 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 $p$.