Solvability of the \(\mathrm{L}^{p}\) Dirichlet problem for the heat equation is equivalent to parabolic uniform rectifiability in the case of a parabolic Lipschitz graph

We prove that if a parabolic Lipschitz (i.e., Lip(1,1/2)) graph domain has the property that its caloric measure is parabolic $A_{\mathrm{\infty }}$ $A_{\infty }$ with respect to surface measure (which property is in turn equivalent to ${\mathrm{L}}^p$ $\mathrm{L}^{p}$ solvability of the Dirichlet problem for some finite $p$ $p$ ), 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 $A_{\mathrm{\infty }}$ $A_{\infty }$ 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 ${\mathrm{L}}^p$ $\mathrm{L}^{p}$ solvability (for some $p<\mathrm{\infty }$ $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.