On the Bakry Emery Condition, the Gradient Estimates and the Local to Global Property of \mathsf{RCD}(K,N)$RCD*(K,N) Metric Measure Spaces
We prove higher summability and regularity of \Gamma \big (f\big $G(f) for functions $f$$f in spaces satisfying the Bakry Emery condition \mathsf{BE}(K,\infty $BE(K, As a byproduct, we obtain various equivalent weak formulations of \mathsf{BE}(K,N)$BE(K,N) and we prove the Local to Global property of the \mathsf{RCD}(K,N)$RCD*(K,N) condition in locally compact metric measure spaces (X,\mathsf{d}\mathfrak m)(X,d,m) without assuming a priori the non branching condition on the metric space.