J. Math. Pures Appl. 105 (2016) 265–292 Contents lists available at ScienceDirect Journal de Mathématiques Pures et Appliquées www.elsevier.com/locate/matpur Poincaré-type inequality for Lipschitz continuous vector fields Giovanna Citti a,* , Maria Manfredini a , Andrea Pinamonti b , Francesco Serra Cassano c a Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, 40126, Bologna, Italy b Scuola Normale Superiore di Pisa, Piazza dei Cavalieri, 7, 56126, Pisa, Italy c Dipartimento di Matematica, Università di Trento Via Sommarive 14, 38123, Povo (Trento), Italy a r t i c l e i n f o a b s t r a c t Article history: Received 23 June 2013 Available online 8 September 2015 MSC: 43A80 35R45 42B20 Keywords: Poincaré inequalities Nonlinear vector fields Lie groups The scope of this paper is to prove a Poincaré type inequality for a family of nonlinear vector fields, whose coefficients are only Lipschitz continuous with respect to the distance induced by the vector fields themselves. © 2015 Elsevier Masson SAS. All rights reserved. r é s u m é Dans cet article on démontre une inégalité de type Poincaré pour une famille de champs de vecteurs non linéaires dont les coefficients sont seulement continus lipschitziens par rapport à la distance induite par les champs eux-mêmes. © 2015 Elsevier Masson SAS. All rights reserved. 1. Introduction and statement of the result The Poincaré inequality is one of the main tools in the proof of regularity of solutions of PDEs in divergence form. Indeed, as proved by Saloff-Coste in [1] and Grigor’yan in [2] (see also [3]), it is equivalent to the Harnack inequality and to Hölder continuity for solutions. Thus, to prove regularity of solutions, it suffices to establish a suitable Poincaré inequality. The Poincaré inequality for smooth Hörmander vector fields is well known and was proved by Jerison [4]. We recall that a Hörmander family of vector fields in R n , is defined by m ≤ n smooth vector fields, say ∇ =(∇ 1 , ..., ∇ m ), such that the generated Lie algebra has maximum rank at every point. Denote by B r (x) ⊂ R n the metric ball of center x and radius r> 0 associated to the CC-distance defined in terms of the family ∇. The Poincaré inequality proved in [4] is: * Corresponding author. E-mail addresses: giovanna.citti@unibo.it (G. Citti), maria.manfredini@unibo.it (M. Manfredini), andrea.pinamonti@gmail.com (A. Pinamonti), casssano@science.unitn.it (F. Serra Cassano). http://dx.doi.org/10.1016/j.matpur.2015.09.001 0021-7824/© 2015 Elsevier Masson SAS. All rights reserved.