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

Abstract

We show if $\partial Ω$ is the graph of a parabolic Lipschitz function, then parabolic surface measure $σ$ of $\partial Ω$ is absolutely continuous with respect to its caloric measure if and only if a (square) Dini-$β$ condition is satisfied. More specifically, the (square) Dini-$β$ condition is that \[\int_0^1 \hatβ(X,t,r)^2 \frac{dr}{r} < \infty, \quad \text{$σ$-a.e. } (X,t) \in \partial Ω.\] Here $\hatβ$ is a parabolic version of the Jones ($L^2$) $β$-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 \[\int_0^1 \hatβ(X,t,r)^2 \frac{dr}{r} < \infty\] up to a set of caloric measure zero then the caloric measure is absolutely continuous with respect to surface measure.