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

  • Nicola Gigli

    SISSA, Trieste, Italy
  • Luca Tamanini

    Universität Bonn, Germany
Second order differentiation formula on RCD*$(K,N)$ spaces cover
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Abstract

The aim of this paper is to prove a second order differentiation formula for H2,2H^{2,2} functions along geodesics in RCD*(K,N)(K,N) spaces with KıinRK ıin \mathbb R and N<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 W2W_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 L2L^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.

Cite this article

Nicola Gigli, Luca Tamanini, Second order differentiation formula on RCD*(K,N)(K,N) spaces. J. Eur. Math. Soc. 23 (2021), no. 5, pp. 1727–1795

DOI 10.4171/JEMS/1042