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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
Published electronically:
September 16, 2009
MathSciNet review:
2557174
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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
 ,  , preprint, to appear in Crelle's Journal.
- 2.
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Franchi and Ermanno
Lanconelli, Hölder regularity theorem for a class of linear
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Scuola Norm. Sup. Pisa Cl. Sci. (4) 10 (1983), no. 4,
523–541. MR
753153 (85k:35094)
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Bruno
Franchi, Guozhen
Lu, and Richard
L. Wheeden, A relationship between Poincaré-type
inequalities and representation formulas in spaces of homogeneous
type, Internat. Math. Res. Notices 1 (1996),
1–14. MR
1383947 (97k:26012), http://dx.doi.org/10.1155/S1073792896000013
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Nicola
Garofalo and Duy-Minh
Nhieu, Isoperimetric and Sobolev inequalities for
Carnot-Carathéodory spaces and the existence of minimal
surfaces, Comm. Pure Appl. Math. 49 (1996),
no. 10, 1081–1144. MR 1404326
(97i:58032), http://dx.doi.org/10.1002/(SICI)1097-0312(199610)49:10<1081::AID-CPA3>3.0.CO;2-A
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Hajłasz and Pekka
Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc.
145 (2000), no. 688, x+101. MR 1683160
(2000j:46063), http://dx.doi.org/10.1090/memo/0688
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Hörmander, Hypoelliptic second order differential
equations, Acta Math. 119 (1967), 147–171. MR 0222474
(36 #5526)
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David
Jerison, The Poincaré inequality for vector fields
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Ermanno
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Morbidelli, On the Poincaré inequality for vector
fields, Ark. Mat. 38 (2000), no. 2,
327–342. MR 1785405
(2002a:46037), http://dx.doi.org/10.1007/BF02384323
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Maheux and L.
Saloff-Coste, Analyse sur les boules d’un opérateur
sous-elliptique, Math. Ann. 303 (1995), no. 4,
713–740 (French). MR 1359957
(96m:35049), http://dx.doi.org/10.1007/BF01461013
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Montanari and Daniele
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Poincaré inequality, Ann. Inst. Fourier (Grenoble)
54 (2004), no. 2, 431–452 (English, with
English and French summaries). MR 2073841
(2005e:46053)
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Daniele
Morbidelli, Fractional Sobolev norms and structure of
Carnot-Carathéodory balls for Hörmander vector fields,
Studia Math. 139 (2000), no. 3, 213–244. MR 1762582
(2002a:46039)
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103–147. MR
793239 (86k:46049), http://dx.doi.org/10.1007/BF02392539
- 13.
Cristian
Rios, Eric
T. Sawyer, and Richard
L. Wheeden, Regularity of subelliptic Monge-Ampère
equations, Adv. Math. 217 (2008), no. 3,
967–1026. MR 2383892
(2009k:35086), http://dx.doi.org/10.1016/j.aim.2007.07.004
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L.
Saloff-Coste, A note on Poincaré, Sobolev, and Harnack
inequalities, Internat. Math. Res. Notices 2 (1992),
27–38. MR
1150597 (93d:58158), http://dx.doi.org/10.1155/S1073792892000047
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B. Street, Multi-parameter Carnot-Carathéodory balls and the theorem of Frobenius, preprint.
- 16.
Terence
Tao and James
Wright, 𝐿^{𝑝} improving bounds
for averages along curves, J. Amer. Math.
Soc. 16 (2003), no. 3, 605–638. MR 1969206
(2004j:42005), http://dx.doi.org/10.1090/S0894-0347-03-00420-X
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Chao-Jiang
Xu and Claude
Zuily, Higher interior regularity for quasilinear subelliptic
systems, Calc. Var. Partial Differential Equations 5
(1997), no. 4, 323–343. MR 1450714
(98e:35039), http://dx.doi.org/10.1007/s005260050069
- 1.
- L. Capogna, G. Citti, M. Manfredini, Smoothness of Lipschitz minimal intrinsic graphs in Heisenberg groups , , 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)
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Additional Information
Maria Manfredini
Affiliation:
Dipartimento di Matematica, Università di Bologna, Piazza Porta S. Donato 5, 40126 Bologna, Italy
Email:
manfredi@dm.unibo.it
DOI:
http://dx.doi.org/10.1090/S0002-9939-09-10054-0
PII:
S 0002-9939(09)10054-0
Received by editor(s):
August 30, 2008
Published electronically:
September 16, 2009
Communicated by:
Matthew J. Gursky
Article copyright:
© Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
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