Exact d'Alembertian for Lorentz distance functions

We refine a recent distributional notion of d'Alembertian of a signed Lorentz distance function to an achronal set in a metric measure spacetime obeying the timelike measure contraction property. We show precise representation formulas and comparison estimates (both upper and lower bounds). Under a condition we call "infinitesimally strict concavity" (known for infinitesimally Minkowskian structures and established here for Finsler spacetimes), we prove the associated distribution is a signed measure certifying the integration by parts formula. This treatment of the d'Alembertian using techniques from metric geometry expands upon its recent nonlinear yet elliptic interpretation; even in the smooth case, our formulas seem to pioneer its exact shape across the timelike cut locus. Two central ingredients our contribution unifies are the localization paradigm of Cavalletti-Mondino and the Sobolev calculus of Beran-Braun-Calisti-Gigli-McCann-Ohanyan-Rott-S\"amann. In the second part of our work, we present several applications of these insights. First, we show the equivalence of the timelike curvature-dimension condition with a Bochner-type inequality. Second, we set up synthetic mean curvature (as well as barriers for CMC sets) exactly. Third, we prove synthetic volume and area estimates of Heintze-Karcher-type, which enable us to show several synthetic volume singularity theorems.