ForareasonableclassofweakPoincaréinequalities,thedecayofthecorrespondingMarkovsemigroupsobtainedearlierbyRcknerandthefirstnamedauthorisimprovedbyremovinganextraL~2-norm.Next,aconcentrationestimateofthereferencemeasureispresentedfortheweakPoincaréinequality,whichissharpasillustratedbysomeexamplesofone-dimensionaldiffusionprocesses.