On the $W$-entropy and Shannon entropy power on RCD$(K, N)$ and RCD$(K, n, N)$ spaces

In this paper, we prove the $W$-entropy formula and the monotonicity and rigidity theorem of the $W$-entropy for the heat flow on RCD$(K, N)$ and RCD$(K, n, N)$ spaces $(X, d, \mu)$, where $K\in \mathbb{R}$, $n\in \mathbb{N}$ is the geometric dimension of $(X, d, \mu)$ and $N\geq n$. We also prove the $K$-concavity of the Shannon entropy power on RCD$(K, N)$ spaces. As an application, we derive the Shannon entropy isoperimetric inequality and the Stam type logarithmic Sobolev inequality on RCD$(0, N)$ spaces with maximal volume growth condition. Finally, we prove the rigidity theorem for the Stam type logarithmic Sobolev inequality with sharp constant on noncollapsing RCD$(0, N)$ spaces.