Annales Academiĉ Scientiarum Fennicĉ
Mathematica
Volumen 38, 2013, 287-308
UNED, ETS de Ingenieros Industriales, Departamento de Matemática Aplicada
28040-Madrid, Spain; edurand 'at' ind.uned.es
Universidad Complutense de Madrid, Departamento de Análisis Matemático
28040-Madrid, Spain; jaramil 'at' mat.ucm.es
University of Cincinnati, Department of Mathematical Sciences
P.O. Box 210025, Cincinnati, OH 45221-0025, U.S.A.; shanmun 'at' ucmail.uc.edu
Abstract. We study a generalization of classical Poincaré 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-Poincaré type inequality of Franchi, Pérez and Wheeden [FPW], the metric measure space should also support a p-Poincaré inequality for some 1 \le p < \infty, and that under weaker assumptions, the metric measure space supports an \infty-Poincaré inequality in the sense of [DJS].
2010 Mathematics Subject Classification: Primary 31E05, 30L99, 43A85.
Key words: Poincaré inequality, BMO-Poincaré inequality, quasiconvexity, Lipschitz functions, Newtonian functions, thick quasiconvexity.
Reference to this article: E. Durand-Cartagena, J.A. Jaramillo and N. Shanmugalingam: First order Poincaré inequalities in metric measure spaces. Ann. Acad. Sci. Fenn. Math. 38 (2013), 287-308.
doi:10.5186/aasfm.2013.3825
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