  We prove that if a parabolic Lipschitz (i.e., Lip(1,1/2)) graph domain has
the property that its caloric measure is a parabolic $A_\infty$ weight with
respect to surface measure (which in turn is equivalent to $L^p$ solvability of
the Dirichlet problem for some finite $p$), then the function defining the
graph has a half-order time derivative in the space of (parabolic) bounded mean
oscillation. Equivalently, we prove that the $A_\infty$ property of caloric
measure implies, in this case, that the boundary is parabolic uniformly
rectifiable. Consequently, by combining our result with the work of Lewis and
Murray we resolve a long standing open problem in the field by characterizing
those parabolic Lipschitz graph domains for which one has $L^p$ solvability
(for some $p <\infty$) of the Dirichlet problem for the heat equation. The key
idea of our proof is to view the level sets of the Green function as extensions
of the original boundary graph for which we can prove (local) square function
estimates of Littlewood-Paley type.
