## Multi-parameter Carnot-Carathéodory balls and the theorem of Frobenius.(English)Zbl 1222.53036

Problems similar to cases of Euclidean spaces and Riemannian manifolds were recently studied and some results, such as implict function theorems, were obtained. This paper considers similar questions on Carnot-Carathéodory spaces, and gives some conclusions, for instance, the Frobenius Theorem and multi-parameter C-C balls. These results are remarkable and interesting.

### MSC:

 53C17 Sub-Riemannian geometry 58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
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### References:

 [1] Bramanti, M., Brandolini, L. and Pedroni, M.: Basic properties of nonsmooth Hörmander’s vector fields and Poincaré’s inequality. Preprint available at · Zbl 1357.46027 [2] Chevalley, C.: Theory of Lie Groups. I . Princeton Mathematical Series 8 . Princeton University Press, Princeton, NJ, 1946. · Zbl 0063.00842 [3] 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. JSTOR: · Zbl 0671.35017 [4] 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. · Zbl 0821.32011 [5] Dieudonné, J.: Foundations of modern analysis . Pure and Applied Mathematics 10 . Academic Press, New York, 1960. · Zbl 0100.04201 [6] 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. · Zbl 0293.35012 [7] Fefferman, C.L. and Sánchez-Calle, A.: Fundamental solutions for second order subelliptic operators. Ann. of Math. (2) 124 (1986), no. 2, 247-272. JSTOR: · Zbl 0613.35002 [8] Hermann, R.: The differential geometry of foliations. II. J. Math. Mech. 11 (1962), 303-315. · Zbl 0152.20502 [9] Hubbard, J.H. and Hubbard, B.B.: Vector calculus, linear algebra, and differential forms. A unified approach. Prentice Hall, Upper Saddle River, NJ, 1999. · Zbl 0918.00001 [10] Izzo, A.J.: $$C^r$$ convergence of Picard’s successive approximations. Proc. Amer. Math. Soc. 127 (1999), no. 7, 2059-2063. JSTOR: · Zbl 0918.34002 [11] Jessen, B., Marcinkiewicz, J. and Zygmund, A.: Note on the differentiability of multiple integrals. Funda. Math. 25 (1935), 217-234. · Zbl 0012.05901 [12] 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. · Zbl 0634.35017 [13] 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. · Zbl 1014.32031 [14] Lundell, A.T.: A short proof of the Frobenius theorem. Proc. Amer. Math. Soc. 116 (1992), no. 4, 1131-1133. · Zbl 0766.58006 [15] Montanari, A. and Morbidelli, D.: Nonsmooth Hörmander’s vector fields and their control balls. To appear in Trans. Amer. Math. Soc. · Zbl 1296.53060 [16] 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. · Zbl 0857.43012 [17] 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. · Zbl 0974.22007 [18] 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. JSTOR: · Zbl 0667.32016 [19] Nagel, A. and Stein, E.M.: Differentiable control metrics and scaled bump functions. J. Differential Geom. 57 (2001), no. 3, 465-492. · Zbl 1041.58006 [20] Nagel, A. and Stein, E.M.: On the product theory of singular integrals. Rev. Mat. Iberoamericana 20 (2004), no. 2, 531-561. · Zbl 1057.42016 [21] Nagel, A. and Stein, E.M.: The $$\overline\partial_ b$$-complex on decoupled boundaries in $$\mathbbC^n$$. Ann. of Math. (2) 164 (2006), no. 2, 649-713. · Zbl 1126.32031 [22] 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. · Zbl 0578.32044 [23] Rampazzo, F.: Frobenius-type theorems for Lipschitz distributions. J. Differential Equations 243 (2007), no. 2, 270-300. · Zbl 1141.53022 [24] Rothschild, L.P. and Stein, E.M.: Hypoelliptic differential operators and nilpotent groups. Acta Math. 137 (1976), no. 3-4, 247-320. · Zbl 0346.35030 [25] Sánchez-Calle, A.: Fundamental solutions and geometry of the sum of squares of vector fields. Invent. Math. 78 (1984), no. 1, 143-160. · Zbl 0582.58004 [26] Spivak, M.: Calculus on manifolds. A modern approach to classical theorems of advanced calculus . W.A. Benjamin, New York-Amsterdam, 1965. · Zbl 0141.05403 [27] 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. · Zbl 0821.42001 [28] Street, B.: An algebra containing the two-sided convolution operators. Adv. Math. 219 (2008), no. 1, 251-315. · Zbl 1148.43009 [29] Thrall, R.M. and Tornheim, L.: Vector spaces and matrices . John Wiley and Sons, New York, 1957. · Zbl 0077.02002 [30] Tao, T. and Wright, J.: $$L^p$$ improving bounds for averages along curves. J. Amer. Math. Soc. 16 (2003), no. 3, 605-638 (electronic). · Zbl 1080.42007
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