First order Poincaré inequalities in metric measure spaces

Abstract

We study a generalization of classical Poincare inequalities, and study conditions that link such an inequality with the first order calculus of functions in the metric measure space setting when the measure is doubling and the metric is complete. The first order calculus considered in this paper is based on the approach of the upper gradient notion of Heinonen and Koskela [HeKo]. We show that under a Vitali type condition on the BMO-Poincare type inequality of Franchi, Perez and Wheeden [FPW], the metric measure space should also support a p-Poincare inequality for some 1 <= p < infinity, and that under weaker assumptions, the metric measure space supports an infinity-Poincare inequality in the sense of [DJS].