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We prove that if a parabolic Lipschitz (i.e., Lip(1,1/2)) graph domain hasthe property that its caloric measure is a parabolic $A_\infty$ weight withrespect to surface measure (which in turn is equivalent to $L^p$ solvability ofthe Dirichlet problem for some finite $p$), then the function defining thegraph has a half-order time derivative in the space of (parabolic) bounded meanoscillation. Equivalently, we prove that the $A_\infty$ property of caloricmeasure implies, in this case, that the boundary is parabolic uniformlyrectifiable. Consequently, by combining our result with the work of Lewis andMurray we resolve a long standing open problem in the field by characterizingthose parabolic Lipschitz graph domains for which one has $L^p$ solvability(for some $p <\infty$) of the Dirichlet problem for the heat equation. The keyidea of our proof is to view the level sets of the Green function as extensionsof the original boundary graph for which we can prove (local) square functionestimates of Littlewood-Paley type.

Solvability of the $L^p$ Dirichlet problem for the heat equation is  equivalent to parabolic uniform rectifiability in the case of a parabolic  Lipschitz graph