  Inspired by the approach of Ivanisvili and Volberg towards functional
inequalities for probability measures with strictly convex potentials, we
investigate the relationship between curvature bounds in the sense of
Bakry-Emery and local functional inequalities. We will show that not only is
the earlier approach for strictly convex potentials extendable to Markov
semigroups and simplified through use of the $\Gamma$-calculus, providing a
consolidating machinery for obtaining functional inequalities new and old in
this general setting, but that a converse also holds. Local inequalities
obtained are proven equivalent to Bakry-Emery curvature. Moreover we will
develop this technique for metric measure spaces satisfying the RCD condition,
providing a unified approach to functional and isoperimetric inequalities in
non-smooth spaces with a synthetic Ricci curvature bound. Finally, we are
interested in commutation properties for semi-group operators on $\mathbb{R}^n$
in the absence of positive curvature, based on a local eigenvalue criteria.
