Nonsmooth Hörmander vector fields and their control balls
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- by Annamaria Montanari and Daniele Morbidelli
- Trans. Amer. Math. Soc. 364 (2012), 2339-2375
- DOI: https://doi.org/10.1090/S0002-9947-2011-05395-X
- Published electronically: December 15, 2011
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Abstract:
We prove a ball-box theorem for nonsmooth Hörmander vector fields of step $s\geq 2.$References
- M. Bramanti, L. Brandolini, M. Pedroni, Basic properties of nonsmooth Hörmander’s vector fields and Poincaré’s inequality, preprint arXiv:0809.2872v2.
- 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. MR 1895205, DOI 10.1090/S0002-9947-02-02928-8
- C. Fefferman and D. H. Phong, Subelliptic eigenvalue problems, Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981) Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983, pp. 590–606. MR 730094
- 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
- 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
- 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
- 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, DOI 10.1007/BF02819446
- Philip Hartman, Ordinary differential equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR 0171038
- Lars Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147–171. MR 222474, DOI 10.1007/BF02392081
- David Jerison, The Poincaré inequality for vector fields satisfying Hörmander’s condition, Duke Math. J. 53 (1986), no. 2, 503–523. MR 850547, DOI 10.1215/S0012-7094-86-05329-9
- Ermanno Lanconelli and Daniele Morbidelli, On the Poincaré inequality for vector fields, Ark. Mat. 38 (2000), no. 2, 327–342. MR 1785405, DOI 10.1007/BF02384323
- 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
- Roberto Monti and Daniele Morbidelli, Trace theorems for vector fields, Math. Z. 239 (2002), no. 4, 747–776. MR 1902060, DOI 10.1007/s002090100342
- 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, DOI 10.4064/sm-139-3-213-244
- 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, DOI 10.1007/BF02392539
- 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.
- 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, DOI 10.1016/j.jde.2006.04.016
- Linda Preiss Rothschild and E. M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math. 137 (1976), no. 3-4, 247–320. MR 436223, DOI 10.1007/BF02392419
- 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, DOI 10.1090/memo/0847
- 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, DOI 10.1007/BF01587936
- 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, DOI 10.1007/978-3-0346-0422-2_{1}7
- B. Street, Multi-parameter Carnot-Carathéodory balls and the theorem of Frobenius, preprint arXiv:0901.2910v3.
- 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, DOI 10.1090/S0894-0347-03-00420-X
Bibliographic 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
- Received by editor(s): December 15, 2008
- Received by editor(s) in revised form: March 5, 2010
- Published electronically: December 15, 2011
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 364 (2012), 2339-2375
- MSC (2010): Primary 53C17; Secondary 35R03
- DOI: https://doi.org/10.1090/S0002-9947-2011-05395-X
- MathSciNet review: 2888209