Second order differentiation formula on RCD*$(K,N)$ spaces

Abstract

The aim of this paper is to prove a second order differentiation formula for $H^{2,2}$ functions along geodesics in RCD*$(K,N)$ spaces with $Kıin\mathbb{R}$ and $N<\infty$. This formula is new even in the context of Alexandrov spaces, where second order differentiation is typically related to semiconvexity.

We establish this result by showing that $W_2$-geodesics can be approximated up to second order, in a sense which we shall make precise, by entropic interpolations. In turn this is achieved by proving new, even in the smooth setting, estimates concerning entropic interpolations which we believe are interesting on their own. In particular we obtain:

• equiboundedness of densities along entropic interpolations,

• local equi-Lipschitz continuity of Schrödinger potentials,

• uniform weighted $L^2$ control of the Hessian of such potentials.

Finally, the techniques adopted in this paper can be used to show that in the RCD setting the viscous solution of the Hamilton–Jacobi equation can be obtained via a vanishing viscosity method, as in the smooth case.

With respect to a previous version, where the space was assumed to be compact, in this paper the second order differentiation formula is proved in full generality.