Inspired by the theory of Sobolev functions on metric measure spaces, we study time functions on measured Lorentzian length spaces. Via the notion of weak lower differential, we are able to define a Sobolev class of such functions. With the help of this machinery, we are able to prove d'Alembert comparison results in the synthetic setting.
This is an ongoing joint project with Tobias Beran, Mathias Braun, Matteo Calisti, Nicola Gigli, Robert McCann, Clemens Sämann and Felix Rott.