A note on the Poincaré inequality for Lipschitz vector fields of step two

Manfredini, Maria

doi:10.1090/S0002-9939-09-10054-0

A note on the Poincaré inequality for Lipschitz vector fields of step two

Author: Maria Manfredini
Journal: Proc. Amer. Math. Soc. 138 (2010), 567-575
MSC (2000): Primary 35A08, 35H20, 43A80; Secondary 35A17, 35J70
DOI: https://doi.org/10.1090/S0002-9939-09-10054-0
Published electronically: September 16, 2009
MathSciNet review: 2557174
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We provide a Poincaré inequality for families of Lipschitz continuous vector fields satisfying a Hörmander-type condition of step two.

1. L. Capogna, G. Citti, M. Manfredini, Smoothness of Lipschitz minimal intrinsic graphs in Heisenberg groups $\mathbb{H}^n$ , , preprint, to appear in Crelle's Journal.
2. B. Franchi, E. Lanconelli, Hölder regularity theorem for a class of linear nonuniformly elliptic operators with measurable coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 10 (1983), 523-541. MR 753153 (85k:35094)
3. B. Franchi, G. Lu, R. Wheeden, A relationship between Poincaré type inequalities and representation formulas in spaces of homogeneous type, Internat. Math. Res. Notices, no. 1, (1996), 1-14. MR 1383947 (97k:26012)
4. N. Garofalo, D.-M. Nhieu, Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces, Comm. Pure Appl. Math. 49 (1996), 1081-1144. MR 1404326 (97i:58032)
5. P. Hajlasz, P. Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc. 145, no. 688 (2000). MR 1683160 (2000j:46063)
6. H. Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147-171. MR 0222474 (36:5526)
7. D. Jerison, The Poincaré inequality for vector fields satisfying Hörmander's condition, Duke Math. J. 53 (1986), 503-523. MR 850547 (87i:35027)
8. E. Lanconelli, D. Morbidelli, On the Poincaré inequality for vector fields, Ark. Mat. 38 (2000), 327-342. MR 1785405 (2002a:46037)
9. P. Maheux, L. Saloff-Coste, Analyse sur les boules d'un opérateur sous-elliptique, Math. Ann. 303 (1995), 713-740. MR 1359957 (96m:35049)
10. A. Montanari, D. Morbidelli, Balls defined by nonsmooth vector fields and the Poincaré inequality, Ann. Inst. Fourier (Grenoble) 54, no. 2 (2004), 431-452. MR 2073841 (2005e:46053)
11. D. Morbidelli, Fractional Sobolev norms and structure of the Carnot-Carathéodory balls for Hörmander vector fields, Studia Math. 139 (2000), 213-244. MR 1762582 (2002a:46039)
12. A. Nagel, E. M. Stein, S. Wainger, Balls and metrics defined by vector fields. I: Basic properties, Acta Math. 155 (1985), 103-147. MR 793239 (86k:46049)
13. C. Rios, E. T. Sawyer, R. L. Wheeden, Regularity of subelliptic Monge-Ampère equations, Adv. Math. 217, no. 3 (2008), 967-1026. MR 2383892
14. L. Saloff-Coste, A note on Poincaré, Sobolev and Harnack inequalities, Internat. Math. Res. Notices, no. 2 (1992), 27-38. MR 1150597 (93d:58158)
15. B. Street, Multi-parameter Carnot-Carathéodory balls and the theorem of Frobenius, preprint.
16. T. Tao, J. Wright, improving bounds for averages along curves, J. Amer. Math. Soc. 16 no. 3 (2003), 605-638. MR 1969206 (2004j:42005)
17. C.-J. Xu, C. Zuily, Higher interior regularity for quasilinear subelliptic systems, Calc. Var. Partial Differential Equations 5 (1997), 323-343. MR 1450714 (98e:35039)