# On the Bakry-Èmery condition, the gradient estimates and the Local-to-Global property of $RCD^*(K,N)$ metric measure spaces

## Ambrosio Luigi - Mondino Andrea - Savaré Giuseppe

accepted
year: 2014
journal: Journal of Geometric Analysis
abstract: We prove higher summability and regularity of $\Gamma(f)$ for functions $f$ in spaces satisfying the Bakry-Èmery condition $BE(K,\infty)$. As a byproduct, we obtain various equivalent weak formulations of $BE(K,N)$ and we prove the Local-to-Global property of the $RCD^*(K,N)$ condition in locally compact metric measure spaces $(X,d,m)$, without assuming a priori the non-branching condition on the metric space.

The paper is available on the cvgmt preprint server.