  We prove that on a large family of metric measure spaces, if the
$L^p$-gradient estimate for heat flows holds for some $p>2$, then the
$L^1$-gradient estimate also holds. This result extends Savar\'e's result on
metric measure spaces, and provides a new proof to von Renesse-Sturm theorem on
smooth metric measure spaces. As a consequence, we propose a new analysis
object based on Gigli's measure-valued Ricci tensor, to characterize the Ricci
curvature of ${\rm RCD}$ space in a local way. The argument is a new iteration
technique based on non-smooth Bakry-\'Emery theory, which is a new method to
study the curvature dimension condition of metric measure spaces.
