The parabolic Dini-$\beta$ condition and absolute continuity of surface and caloric measure

We show if $\mathrm{\partial }\mathrm{\Omega }$ is the graph of a parabolic Lipschitz function, then parabolic surface measure $\sigma$ of $\mathrm{\partial }\mathrm{\Omega }$ is absolutely continuous with respect to its caloric measure if and only if a (square) Dini-$\beta$ condition is satisfied. More specifically, the (square) Dini-$\beta$ condition is that $${\int }_0^1\widehat{\beta }(X,t,r{)}^2\frac{dr}{r}<\mathrm{\infty },\sigma \text{-a.e. }(X,t)\in \mathrm{\partial }\mathrm{\Omega }\mathrm{.}$$ Here $\widehat{\beta }$ is a parabolic version of the Jones ($L^2$) $\beta$-numbers. We show that these conditions are satisfied if and only if the graph is covered by a countable collection of {\it regular} Lipschitz graphs, that is, graphs with additional in-time regularity in the form of a half order time derivative in the parabolic BMO space. This supports the view that covering by {\it regular} Lipschitz graphs is the right notion for qualitative parabolic rectifiability in the context of parabolic PDEs. We also show that if $${\int }_0^1\widehat{\beta }(X,t,r{)}^2\frac{dr}{r}<\mathrm{\infty }$$ up to a set of caloric measure zero then the caloric measure is absolutely continuous with respect to surface measure.