Revista Matemática Iberoamericana

Multi-parameter Carnot-Carathéodory balls and the theorem of Frobenius

Brian Street

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Abstract

We study multi-parameter Carnot-Carathéodory balls, generalizing results due to Nagel, Stein and Wainger in the single parameter setting. The main technical result is seen as a uniform version of the theorem of Frobenius. In addition, we study maximal functions associated to certain multi-parameter families of Carnot-Carathéodory balls.

Article information

Source
Rev. Mat. Iberoamericana, Volume 27, Number 2 (2011), 645-732.

Dates
First available in Project Euclid: 10 June 2011

Permanent link to this document
https://projecteuclid.org/euclid.rmi/1307713041

Mathematical Reviews number (MathSciNet)
MR2848534

Zentralblatt MATH identifier
1222.53036

Subjects
Primary: 53C17: Sub-Riemannian geometry
Secondary: 53C12: Foliations (differential geometric aspects) [See also 57R30, 57R32] 42B25: Maximal functions, Littlewood-Paley theory

Keywords
sub-Riemannian geometry Carnot-Carathéodory geometry Frobenius theorem multi-parameter spaces of homogeneous type maximal functions

Citation

Street, Brian. Multi-parameter Carnot-Carathéodory balls and the theorem of Frobenius. Rev. Mat. Iberoamericana 27 (2011), no. 2, 645--732. https://projecteuclid.org/euclid.rmi/1307713041


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References

  • Bramanti, M., Brandolini, L. and Pedroni, M.: Basic properties of nonsmooth Hörmander's vector fields and Poincaré's inequality. Preprint available at arXiv:0809.2872, 2008.
  • Chevalley, C.: Theory of Lie Groups. I. Princeton Mathematical Series 8. Princeton University Press, Princeton, NJ, 1946.
  • Christ, M.: Regularity properties of the $\overline\partial_b$ equation on weakly pseudoconvex CR manifolds of dimension 3. J. Amer. Math. Soc. 1 (1988), no. 3, 587-646.
  • Chang, D.-C., Nagel, A. and Stein, E.M.: Estimates for the $\overline\partial$-Neumann problem in pseudoconvex domains of finite type in $\mathbbC^2$. Acta Math. 169 (1992), no. 3-4, 153-228.
  • Dieudonné, J.: Foundations of modern analysis. Pure and Applied Mathematics 10. Academic Press, New York, 1960.
  • Folland, G.B. and Stein, E.M.: Estimates for the $\bar\partial_b$ complex and analysis on the Heisenberg group. Comm. Pure Appl. Math. 27 (1974), 429-522.
  • Fefferman, C.L. and Sánchez-Calle, A.: Fundamental solutions for second order subelliptic operators. Ann. of Math. (2) 124 (1986), no. 2, 247-272.
  • Hermann, R.: The differential geometry of foliations. II. J. Math. Mech. 11 (1962), 303-315.
  • Hubbard, J.H. and Hubbard, B.B.: Vector calculus, linear algebra, and differential forms. A unified approach. Prentice Hall, Upper Saddle River, NJ, 1999.
  • Izzo, A.J.: $C^r$ convergence of Picard's successive approximations. Proc. Amer. Math. Soc. 127 (1999), no. 7, 2059-2063.
  • Jessen, B., Marcinkiewicz, J. and Zygmund, A.: Note on the differentiability of multiple integrals. Funda. Math. 25 (1935), 217-234.
  • Jerison, D. and Sánchez-Calle, A.: Subelliptic, second order differential operators. In Complex analysis, III (College Park, Md., 1985-86), 46-77. Lecture Notes in Math. 1277. Springer, Berlin, 1987.
  • Koenig, K.D.: On maximal Sobolev and Hölder estimates for the tangential Cauchy-Riemann operator and boundary Laplacian. Amer. J. Math. 124 (2002), no. 1, 129-197.
  • Lundell, A.T.: A short proof of the Frobenius theorem. Proc. Amer. Math. Soc. 116 (1992), no. 4, 1131-1133.
  • Montanari, A. and Morbidelli, D.: Nonsmooth Hörmander's vector fields and their control balls. To appear in Trans. Amer. Math. Soc.
  • Müller, D., Ricci, F. and Stein, E.M.: Marcinkiewicz multipliers and multi-parameter structure on Heisenberg (-type) groups. I. Invent. Math. 119 (1995), no. 2, 199-233.
  • Nagel, A., Ricci, F. and Stein, E.M.: Singular integrals with flag kernels and analysis on quadratic CR manifolds. J. Funct. Anal. 181 (2001), no. 1, 29-118.
  • Nagel, A., Rosay, J.-P., Stein, E.M. and Wainger, S.: Estimates for the Bergman and Szegő kernels in $\textbfC^2$. Ann. of Math. (2) 129 (1989), no. 1, 113-149.
  • Nagel, A. and Stein, E.M.: Differentiable control metrics and scaled bump functions. J. Differential Geom. 57 (2001), no. 3, 465-492.
  • Nagel, A. and Stein, E.M.: On the product theory of singular integrals. Rev. Mat. Iberoamericana 20 (2004), no. 2, 531-561.
  • Nagel, A. and Stein, E.M.: The $\overline\partial\sb b$-complex on decoupled boundaries in $\mathbbC^n$. Ann. of Math. (2) 164 (2006), no. 2, 649-713.
  • Nagel, A., Stein, E.M. and Wainger, S.: Balls and metrics defined by vector fields. I. Basic properties. Acta Math. 155 (1985), no. 1-2, 103-147.
  • Rampazzo, F.: Frobenius-type theorems for Lipschitz distributions. J. Differential Equations 243 (2007), no. 2, 270-300.
  • Rothschild, L.P. and Stein, E.M.: Hypoelliptic differential operators and nilpotent groups. Acta Math. 137 (1976), no. 3-4, 247-320.
  • Sánchez-Calle, A.: Fundamental solutions and geometry of the sum of squares of vector fields. Invent. Math. 78 (1984), no. 1, 143-160.
  • Spivak, M.: Calculus on manifolds. A modern approach to classical theorems of advanced calculus. W.A. Benjamin, New York-Amsterdam, 1965.
  • Stein, E.M.: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton Mathematical Series 43. Monographs in Harmonic Analysis, III. Princeton University Press, Princeton, NJ, 1993.
  • Street, B.: An algebra containing the two-sided convolution operators. Adv. Math. 219 (2008), no. 1, 251-315.
  • Thrall, R.M. and Tornheim, L.: Vector spaces and matrices. John Wiley and Sons, New York, 1957.
  • Tao, T. and Wright, J.: $L^p$ improving bounds for averages along curves. J. Amer. Math. Soc. 16 (2003), no. 3, 605-638 (electronic).