*preprint*

**Inserted:** 22 oct 2017

**Last Updated:** 22 oct 2017

**Year:** 2017

**Abstract:**

We prove that on a large family of metric measure spaces, if the `carre du champ' $\Gamma$ satisfies a $p$-gradient estimate of heat flows for some $p>2$, then the metric measure space is ${\rm RCD}(K,\infty)$. The argument relies on the non-smooth Bakry-Emery's theory. As an application, we provide another proof of the von Renesse-Sturm's theorem on smooth metric measure space.

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