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

## TL;DR

This paper establishes that synthetic lower Ricci bounds in time-dependent metric measure spaces can be characterized by various functional inequalities, providing a new framework for understanding super-Ricci flows.

## Contribution

It extends the characterization of Ricci bounds via functional inequalities to the setting of time-dependent metric measure spaces and super-Ricci flows.

## Key findings

- Equivalence of Ricci bounds and functional inequalities in static spaces
- Extension of these equivalences to time-dependent spaces
- Characterization of super-Ricci flows through these inequalities

## Abstract

We prove that synthetic lower Ricci bounds for metric measure spaces -- both in the sense of Bakry-\'Emery and in the sense of Lott-Sturm-Villani -- can be characterized by various functional inequalities including local Poincar\'e 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.