A Lorentzian Lasry-Lions regularization theorem

The main goal of this paper is to establish a general Lorentzian Lasry-Lions regularization theorem: let $u$ be a function defined on a globally hyperbolic spacetime. Assume that its forward Lax–Oleinik evolution $Tu$ is locally semiconcave in a neighbourhood of $(t_0,y_0)$ and has future-directed timelike superdifferentials there. Then, for $t$ close to $t_0$ and sufficiently small $s>0$, the function ${\widehat{T}}_s\circ T_{t}u$ is of class $C_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,1}$ in a neighbourhood of $y_0$. We give sufficient conditions ensuring the assumptions of the theorem and present an application to optimal transport: under quite general assumptions, for any two intermediate measures along a displacement interpolation, there exists a $C_{loc}^{1,1}$-regular maximizing pair in the dual formulation.