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We prove that synthetic lower Ricci bounds for metric measure spaces — both in the sense of Bakry–Émery and in the sense of Lott–Sturm–Villani — can be characterized by various functional inequalities including local Poincaré inequalities, local logarithmic Sobolev inequalities, dimension independent Harnack inequality, and logarithmic Harnack inequality.  More generally, these equivalences will be proven in the setting of time‐dependent metric measure spaces and will provide a characterization of super‐Ricci flows of metric measure spaces.

Functional inequalities for the heat flow on time‐dependent metric measure spaces