Nonsmooth Hörmander vector fields and their control balls

Montanari, Annamaria; Morbidelli, Daniele

doi:10.1090/S0002-9947-2011-05395-X

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
DOI: https://doi.org/10.1090/S0002-9947-2011-05395-X
Published electronically: December 15, 2011
MathSciNet review: 2888209
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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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/10.1007/978-3-0346-0422-2_17

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 https://doi.org/10.1090/S0894-0347-03-00420-X