|
Nonsmooth Hörmander vector fields and their control balls
Authors:
Annamaria Montanari and Daniele Morbidelli
Journal:
Trans. Amer. Math. Soc. 364 (2012), 2339-2375
MSC (2010):
Primary 53C17; Secondary 35R03
Posted:
December 15, 2011
Full-text PDF
Abstract |
References |
Similar Articles |
Additional Information
Abstract: We prove a ball-box theorem for nonsmooth Hörmander vector fields of step
References
- [BBP]
M. Bramanti, L. Brandolini, M. Pedroni, Basic properties of nonsmooth Hörmander's vector fields and Poincaré's inequality, preprint arXiv:0809.2872v2.
- [CM]
G.
Citti and A.
Montanari, Regularity properties of solutions of
a class of elliptic-parabolic nonlinear Levi type equations, Trans. Amer. Math. Soc. 354 (2002), no. 7, 2819–2848 (electronic).
MR
1895205 (2004k:35276), http://dx.doi.org/10.1090/S0002-9947-02-02928-8
- [FP]
C.
Fefferman and D.
H. Phong, Subelliptic eigenvalue problems, (Chicago, Ill.,
1981) Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983,
pp. 590–606. MR 730094
(86c:35112)
- [FL1]
Bruno
Franchi and Ermanno
Lanconelli, Une métrique associée à une classe
d’opérateurs elliptiques
dégénérés, Rend. Sem. Mat. Univ. Politec.
Torino Special Issue (1983), 105–114 (1984)
(French). Conference on linear partial and pseudodifferential operators
(Torino, 1982). MR 745979
(86d:35057)
- [FL2]
Bruno
Franchi and Ermanno
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), no. 4,
523–541. MR
753153 (85k:35094)
- [FSSC]
Bruno
Franchi, Raul
Serapioni, and Francesco
Serra Cassano, Approximation and imbedding theorems for weighted
Sobolev spaces associated with Lipschitz continuous vector fields,
Boll. Un. Mat. Ital. B (7) 11 (1997), no. 1,
83–117 (English, with Italian summary). MR 1448000
(98c:46062)
- [GN]
Nicola
Garofalo and Duy-Minh
Nhieu, Lipschitz continuity, global smooth approximations and
extension theorems for Sobolev functions in Carnot-Carathéodory
spaces, J. Anal. Math. 74 (1998), 67–97. MR 1631642
(2000i:46025), http://dx.doi.org/10.1007/BF02819446
- [Ha]
Philip
Hartman, Ordinary differential equations, John Wiley &
Sons Inc., New York, 1964. MR 0171038
(30 #1270)
- [H]
Lars
Hörmander, Hypoelliptic second order differential
equations, Acta Math. 119 (1967), 147–171. MR 0222474
(36 #5526)
- [J]
David
Jerison, The Poincaré inequality for vector fields
satisfying Hörmander’s condition, Duke Math. J.
53 (1986), no. 2, 503–523. MR 850547
(87i:35027), http://dx.doi.org/10.1215/S0012-7094-86-05329-9
- [LM]
Ermanno
Lanconelli and Daniele
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
- [MM]
Annamaria
Montanari and Daniele
Morbidelli, Balls defined by nonsmooth vector fields and the
Poincaré inequality, Ann. Inst. Fourier (Grenoble)
54 (2004), no. 2, 431–452 (English, with
English and French summaries). MR 2073841
(2005e:46053)
- [MoM]
Roberto
Monti and Daniele
Morbidelli, Trace theorems for vector fields, Math. Z.
239 (2002), no. 4, 747–776. MR 1902060
(2003c:46046), http://dx.doi.org/10.1007/s002090100342
- [M]
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)
- [NSW]
Alexander
Nagel, Elias
M. Stein, and Stephen
Wainger, Balls and metrics defined by vector fields. I. Basic
properties, Acta Math. 155 (1985), no. 1-2,
103–147. MR
793239 (86k:46049), http://dx.doi.org/10.1007/BF02392539
- [RaS]
F. Rampazzo, H. J. Sussmann, Set-valued differential and a nonsmooth version of Chow's theorem, Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, Florida, 2001.
- [RaS2]
Franco
Rampazzo and Héctor
J. Sussmann, Commutators of flow maps of nonsmooth vector
fields, J. Differential Equations 232 (2007),
no. 1, 134–175. MR 2281192
(2007j:49021), http://dx.doi.org/10.1016/j.jde.2006.04.016
- [RoS]
Linda
Preiss Rothschild and E.
M. Stein, Hypoelliptic differential operators and nilpotent
groups, Acta Math. 137 (1976), no. 3-4,
247–320. MR 0436223
(55 #9171)
- [SW]
Eric
T. Sawyer and Richard
L. Wheeden, Hölder continuity of weak solutions to subelliptic
equations with rough coefficients, Mem. Amer. Math. Soc.
180 (2006), no. 847, x+157. MR 2204824
(2007f:35037)
- [Sem]
S.
Semmes, Finding curves on general spaces through quantitative
topology, with applications to Sobolev and Poincaré
inequalities, Selecta Math. (N.S.) 2 (1996),
no. 2, 155–295. MR 1414889
(97j:46033), http://dx.doi.org/10.1007/BF01587936
- [Ste]
Elias
M. Stein, Some geometrical concepts arising in harmonic
analysis, Geom. Funct. Anal. Special Volume (2000),
434–453. GAFA 2000 (Tel Aviv, 1999). MR 1826263
(2002f:42014), http://dx.doi.org/10.1007/978-3-0346-0422-2_17
- [Str]
B. Street, Multi-parameter Carnot-Carathéodory balls and the theorem of Frobenius, preprint arXiv:0901.2910v3.
- [TW]
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
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC (2010):
53C17,
35R03
Retrieve articles in all journals
with MSC (2010):
53C17,
35R03
Additional Information
Annamaria Montanari
Affiliation:
Dipartimento di Matematica, Università di Bologna, Piazza di Porta S Donato 5, 40127 Bologna, Italy
Email:
montanar@dm.unibo.it
Daniele Morbidelli
Affiliation:
Dipartimento di Matematica, Università di Bologna, Piazza di Porta S Donato 5, 40127 Bologna, Italy
Email:
morbidel@dm.unibo.it
DOI:
http://dx.doi.org/10.1090/S0002-9947-2011-05395-X
PII:
S 0002-9947(2011)05395-X
Received by editor(s):
December 15, 2008
Received by editor(s) in revised form:
March 5, 2010
Posted:
December 15, 2011
Article copyright:
© Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain after
28 years from publication.
|