Hölder a priori estimates for second order tangential operators on CR manifolds
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 2 (2003) no. 2, p. 345-378

On a real hypersurface M in n+1 of class C 2,α we consider a local CR structure by choosing n complex vector fields W j in the complex tangent space. Their real and imaginary parts span a 2n-dimensional subspace of the real tangent space, which has dimension 2n+1. If the Levi matrix of M is different from zero at every point, then we can generate the missing direction. Under this assumption we prove interior a priori estimates of Schauder type for solutions of a class of second order partial differential equations with C α coefficients, which are not elliptic because they involve second-order differentiation only in the directions of the real and imaginary part of the tangential operators W j . In particular, our result applies to a class of fully nonlinear PDE’s naturally arising in the study of domains of holomorphy in the theory of holomorphic functions of several complex variables.

Classification:  35J70,  35H20,  32W50,  22E30
@article{ASNSP_2003_5_2_2_345_0,
     author = {Montanari, Annamaria},
     title = {H\"older a priori estimates for second order tangential operators on CR manifolds},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola normale superiore},
     volume = {Ser. 5, 2},
     number = {2},
     year = {2003},
     pages = {345-378},
     zbl = {1170.35433},
     mrnumber = {2005607},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2003_5_2_2_345_0}
}
Montanari, Annamaria. Hölder a priori estimates for second order tangential operators on CR manifolds. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 2 (2003) no. 2, pp. 345-378. http://www.numdam.org/item/ASNSP_2003_5_2_2_345_0/

[1] E. Bedford - B. Gaveau, Hypersurfaces with Bounded Levi Form, Indiana Univ. J. 27 n. 5 (1978), 867-873. | MR 499287 | Zbl 0365.32011

[2] L. Caffarelli - J. J. Kohn - L. Niremberg - J. Spruck, The Dirichlet problem for non-linear second order elliptic equations II: Complex Monge-Ampère and uniformly elliptic equations, Comm. Pure Appl. Math. 38 (1985), 209-252. | MR 780073 | Zbl 0598.35048

[3] L. Capogna - D. Danielli - N. Garofalo, Capacitary estimates and the local behavior of solutions of nonlinear subelliptic equations., Am. J. Math. 118, n. 6 (1996), 1153-1196. | MR 1420920 | Zbl 0878.35020

[4] G. Citti, C regularity of solutions of a quasilinear equation related to the Levi operator, Ann. Scuola Norm. Sup. di Pisa Cl. Sci., Serie 4 Vol. XXIII (1996), 483-529. | Numdam | MR 1440031 | Zbl 0872.35018

[5] G. Citti, C regularity of solutions of the Levi equation, Ann. Inst. H. Poincare, Anal. non Linéaire 15 n. 4 (1998), 517-534. | Numdam | MR 1632929 | Zbl 0921.35033

[6] G. Citti, Regularity of solutions of a nonlinear Hörmander type equation, Nonlinear Anal. 47 (2001), 479-489. | MR 1970667 | Zbl 1042.35517

[7] G. Citti - E. Lanconelli - A. Montanari, On the smoothness of viscosity solutions of the prescribed Levi-curvature equation, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 10 (1999), 61-68. | MR 1768190 | Zbl 1010.35024

[8] G. Citti - E. Lanconelli - A. Montanari, Smoothness of Lipschitz continuous graphs with non vanishing Levi curvature, Acta Math. 188 (2002), 87-128. | MR 1947459 | Zbl 1030.35084

[9] G. Citti - A. Montanari, Strong solutions for the Levi curvature equation, Adv. Differential Equations 5 (1-3) (2000), 323-342. | MR 1734545 | Zbl pre01700721

[10] G. Citti - A. Montanari, Regularity properties of Levi flat graphs, C.R. Acad. Sci. Paris 329 n. 1 (1999), 1049-1054. | MR 1735882 | Zbl 0940.35053

[11] G. Citti - A. Montanari, Analytic estimates for solutions of the Levi equation, J. Differential Equations 173 (2001), 356-389. | MR 1834119 | Zbl 1098.35517

[12] G. Citti - A. Montanari, C regularity of solutions of an equation of Levi’s type in 2n+1 , Ann. Mat. Pura Appl. 180 (2001), 27-58. | MR 1848050 | Zbl 1030.35019

[13] G. Citti - A. Montanari, Regularity properties of solutions of a class of elliptic-parabolic nonlinear Levi type equations, Trans. Amer. Math. Soc. 354 (2002), 2819-2848. | MR 1895205 | Zbl 1008.35030

[14] J. P. D'Angelo, “Several Complex Variables and the Geometry of Real Hypersurfaces”, Studies in Advanced Mathematics, CRC Press, Boca Raton, Florida, 1993. | MR 1224231 | Zbl 0854.32001

[15] G. B. Folland, Subelliptic estimates and functions spaces on nilpotent Lie groups, Ark. Mat. 13 (1975), 161-207. | MR 494315 | Zbl 0312.35026

[16] G. B. Folland - E. M. Stein, Estimates for the ¯ b complex and analysis on the Heisenberg group, Comm. Pure Appl. Math. 20 (1974), 429-522. | MR 367477 | Zbl 0293.35012

[17] D. Gilgarg - N. S. Trudinger, “Elliptic partial differential equations of second order”, Grundlehrer der Math. Wiss. Vol. 224, Springer-Verlag, New York, 1977. | MR 473443 | Zbl 0361.35003

[18] L. Hörmander, “An Introduction to Complex Analysis in Several Variables”, Von Nostrand, Princeton, NJ, 1966. | MR 203075 | Zbl 0138.06203

[19] L. Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147-171. | MR 222474 | Zbl 0156.10701

[20] S. Krantz, “Function Theory of Several Complex Variables”, Wiley, New York, 1982. | MR 635928 | Zbl 0471.32008

[21] F. Lascialfari, A. Montanari, Smooth regularity for solutions of the Levi Monge-Ampère equation, to appear on Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 12 (2001), 115-123. | MR 1898454 | Zbl 1097.35061

[22] A. Montanari - F. Lascialfari, The Levi Monge-Ampère equation: smooth regularity of strictly Levi convex solutions, preprint. | Zbl pre02147105

[23] A. Nagel - E. M. Stein - S. Wainger, Balls and metrics defined by vector fields I: basic properties, Acta Math. 155 (1985), 103-147. | MR 793239 | Zbl 0578.32044

[24] R. M. Range, “Holomorphic Functions and Integral Representation Formulas in Several Complex Variables, Springer-Verlag, New York, 1986. | MR 847923 | Zbl 0591.32002

[25] L. P. Rothschild - E. M. Stein, Hypoelliptic differential operators on nilpotent groups, Acta Math. 137 (1977), 247-320. | MR 436223 | Zbl 0346.35030

[26] A. Sánchez-Calle, Fundamental solutions and geometry of the sum of squares of vector fields, Invent. Math. 78 (1984), 143-160. | MR 762360 | Zbl 0582.58004

[27] Z. Slodkowski - G. Tomassini, The Levi equation in higher dimension and relationships to the envelope of holomorphy, Amer. J. Math. 116 (1994), 479-499. | MR 1269612 | Zbl 0802.35050

[28] Z. Slodkowski - G. Tomassini, Weak solutions for the Levi equation and Envelope of Holomorphy, J. Funct. Anal. 101, n. 4 (1991), 392-407. | MR 1136942 | Zbl 0744.35015

[29] E. M. Stein, “Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals”, Princeton university Press, Princeton, New Jersey 1993. | MR 1232192 | Zbl 0821.42001

[30] G. Tomassini, Geometric Properties of Solutions of the Levi equation, Ann. Mat. Pura Appl. 152 (4) (1988), 331-344. | MR 980986 | Zbl 0681.35017

[31] C. J. Xu, Regularity for Quasilinear Second-Order Subelliptic Equations, Comm. Pure Appl. Math. 45 (1992), 77-96. | MR 1135924 | Zbl 0827.35023