On the Bakry–Émery Condition, the Gradient Estimates and the Local-to-Global Property of RCD∗(K, N) Metric Measure Spaces

We prove higher summability and regularity of Gamma(f) for functions f in spaces satisfying the Bakry–Émery condition BE(K,∞). 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.