BMO Solvability and Absolute Continuity of Caloric Measure

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 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 Lp solvability of the initial-Dirichlet problem, we may then deduce that BMO-solvability implies Lp solvability for some finite p.