  We prove higher summability and regularity of \Gamma(f) for functions f in
spaces satisfying the Bakry-\'Emery 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.
