$RCD^*(K,N)$空间上的二阶微分公式
Second order differentiation formula on $RCD^*(K,N)$ spaces
摘要：
在具有$N < \infty$的$RCD^*(K,N)$空间中证明$H^{2,2}$函数沿测地线的二阶微分公式.这个公式甚至在亚历山德罗夫空间的上下文中也是新的，在亚历山德罗夫空间中，二阶微分通常与双凸性有关。我们建立这个结果表明，$W_2$ -测地线可以近似到二阶，在某种意义上，我们将作出精确的熵插值。反过来，这是通过证明新的，即使在顺利设置，估计熵插值，我们相信是有趣的自己。特别地，我们获得：- 沿熵插值的密度的等界性，-薛定谔势的局部等Lipschitz连续性，-这种势的Hessian的均匀加权$L^2$控制。最后，本文所采用的技术可以用来表明，在$RCD$设置的Hamilton-Jacobi方程的粘性解可以通过消失的粘性方法，根据光滑的情况下获得。相对于以前的一个版本，其中的空间被假定为是紧凑的，本文证明了二阶微分公式在充分的一般性。
英文摘要：
Aim of this paper is to prove the second order differentiation formula for $H^{2,2}$ functions along geodesics in $RCD^*(K,N)$ spaces with $N < \infty$. 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 interpolation. 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 the densities along the entropic interpolations, - local equi-Lipschitz continuity of the Schrödinger potentials, - a 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, in accordance with 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.
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