Twisting and intertwining

Intertwining relations between semigroups is a useful tool for functional inequalities such as Poincaré or Log-Sobolev. Yet, the usual assumption to obtain this relation, the Bakry-Emery criterion, lacks optimality. The twisted intertwining method brings interesting results in some case where Bakry-Emery criterion does not provide any and can correct the lack of optimality.