Bakry-Emery Conditions on Almost Smooth Metric Measure Spaces

In this short note, we give a sufficient condition for almost smooth compact metric measure spaces to satisfy the Bakry-Emery condition BE(K, N). The suffcient condition is satisfied for the glued space of any two (not necessary same dimensional) closed pointed Riemannian manifolds at their base points. This tells us that the BE condition is strictly weaker than the RCD condition even in this setting, and that the local dimension is not constant even if the space satisfies the BE condition with the coincidence between the induced distance by the Cheeger energy and the original distance. In particular, the glued space gives affrst example with a Ricci bound from below in the Bakry-Emery sense, whose local dimension is not constant. We also give a necessary and suffcient condition for such spaces to be RCD(K, N) spaces.