Abstract
We introduce a new type of first order Poincaré inequality for functions defined on a metric measure space, that is an useful tool in the study of Newtonian spaces based on Banach function spaces. This Poincaré inequality extends the Orlicz-Poincaré inequality introduced by Aïssaoui (2004) and the Poincaré inequality based on Lorentz spaces, introduced by Costea and Miranda (2011), that in turn generalize the well-known weak (1,p)-Poincaré inequality. Using very recent results of Durand-Cartagena, Jaramillo and Shanmugalingam (2012, 2013), it turns out that every complete metric space X, endowed with a doubling measure and supporting a weak Poincaré inequality based on a Banach function space is (thick) quasiconvex. We prove that the validity of the Poincaré inequality based on a Banach function space, on a doubling metric measure space, implies a pointwise estimate involving an appropriate maximal operator.