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Regularity of convex functions on Heisenberg groups. (English) Zbl 1121.43007
The Heisenberg nilpotent Lie groups arise by releasing the quantum–mechanically motivated Heisenberg commutation relations from their operator realizations in Hilbert space [H. Weyl, The Theory of Groups and Quantum Mechanics. Dover, New York (1949; Zbl 0041.56804), A. Borel, Hermann Weyl and Lie groups. Hermann Weyl Centenary Lectures, Berlin: Springer-Verlag (1986; Zbl 0599.01014), pp. 53–82; Collected Papers, Vol. IV, New York: Springer-Verlag (2001; Zbl 1065.01016)]. Apart from their importance for the foundations of string theory [J. Lepowski, Perspectives on vertex operators and the monster. The Mathematical Heritage of Hermann Weyl, Proc. Symp. Pure Math. 48, 181–197 (1988; Zbl 0662.17007), W. Schempp [Indistinguishability and quantum entanglement, to be published in Int. J. Pure Appl. Math.], the unitary representation theory of Heisenberg Lie groups forms a highly interesting topic of mathematical research [A. Weil, Acta Math. 111, 143–211 (1964; Zbl 0203.03305); Collected Papers, Vol. III, New York: Springer-Verlag (1979; Zbl 0424.01029), pp. 1–69, A. Borel, N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups. Annals of Mathematics Studies, 94. Princeton, New Jersey: Princeton University Press (1980; Zbl 0443.22010)].
In the vein of analysis and sub–Riemannian geometry of Heisenberg groups [A. Bellaïche, Prog. Math. 144, 1–78 (1996; Zbl 0862.53031), W. Schempp, Math. Methods Appl. Sci. 22, No. 11, 867–922 (1999; Zbl 0964.92026)], the paper under review is concerned with differentiability properties of convex functions [P. Pansu, Ann. Math. (2) 129, No. 1, 1–60 (1989; Zbl 0678.53042)]. It establishes that the concepts of horizontal convexity and viscosity convexity are equivalent notions [D. Danielli, N. Carofalo and D.-M. Nhieu, Commun. Anal. Geom. 11, No. 2, 263–341 (2003; Zbl 1077.22007)], and that horizontally convex functions are locally Lipschitz continuous with respect to the Carnot-Carathéodory metric [M. Gromov, Prog. Math. 144, 79–323 (1996; Zbl 0864.53025)]. In addition, the article exhibits horizontally convex functions of Weierstraß type which are nowhere differentiable on a dense set of vertical lines.

MSC:
43A80 Analysis on other specific Lie groups
26B25 Convexity of real functions of several variables, generalizations
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
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References:
[1] O. Alvarez - J.-M. Larsy - P.-L. Lions, Convex viscosity solutions and state constraints, J. Math. Pures Appl. (76) 9 (1997), 265-288. Zbl0890.49013 MR1441987 · Zbl 0890.49013 · doi:10.1016/S0021-7824(97)89952-7
[2] L. Ambrosio - N. Fusco - D. Pallara, “Functions of Bounded Variation and Free Discontinuity Problems”, Oxford Mathematical Monographs, Oxford Science Publications, Clarendon Press, Oxford, 2000. Zbl0957.49001 MR1857292 · Zbl 0957.49001
[3] L. Ambrosio - V. Magnani, Weak differentiability of BV functions on stratified groups, Math. Z. (1) 245 (2003), 123-153. Zbl1048.49030 MR2023957 · Zbl 1048.49030 · doi:10.1007/s00209-003-0530-2
[4] Z. Balogh - M. Rickly - F. Serra Cassano, Comparison of Hausdorff measures with respect to the Euclidean and the Heisenberg metric, Publ. Mat. 47 (2003), 237-259. Zbl1060.28002 MR1970902 · Zbl 1060.28002 · doi:10.5565/PUBLMAT_47103_11 · eudml:41487
[5] T. Bieske, On \(\infty \)-harmonic functions on the Heisenberg group, Comm. Partial Differential Equations 27 (2002), 727-762. Zbl1090.35063 MR1900561 · Zbl 1090.35063 · doi:10.1081/PDE-120002872
[6] X. Cabre - L. Caffarelli, Fully nonlinear elliptic equations, AMS colloquium publications 43, AMS, Providence, RI, 1995. Zbl0834.35002 MR1351007 · Zbl 0834.35002
[7] M. Crandall - C. Evans - P.-L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. (2) 282 (1984), 487-502. Zbl0543.35011 MR732102 · Zbl 0543.35011 · doi:10.2307/1999247
[8] M. Crandall - H. Ishii - P.-L. Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (1) 27 (1992), 1-67. Zbl0755.35015 MR1118699 · Zbl 0755.35015 · doi:10.1090/S0273-0979-1992-00266-5
[9] J. Cygan, Subadditivity of homogeneous norms on certain nilpotent Lie groups, Proc. Amer. Math. Soc. 83 (1981), 69-70. Zbl0475.43010 MR619983 · Zbl 0475.43010 · doi:10.2307/2043893
[10] D. Danielli - N. Garofalo - D.-M. Nhieu, Notions of convexity in Carnot groups, Comm. Anal. Geom. (2) 11 (2003), 263-341. Zbl1077.22007 MR2014879 · Zbl 1077.22007 · doi:10.4310/CAG.2003.v11.n2.a5
[11] D. Danielli - N. Garofalo - D.-M. Nhieu - F. Tournier, The theorem of Busemann-Feller-Alexandrov in Carnot groups, preprint. Zbl1071.22004 MR2104079 · Zbl 1071.22004 · doi:10.4310/CAG.2004.v12.n4.a5
[12] L. Evans - R. Gariepy, “Measure Theory and Fine Properties of Functions”, Studies in Advanced Mathematics, CRC Press, 1992. Zbl0804.28001 MR1158660 · Zbl 0804.28001
[13] K. J. Falconer, “Fractal Geometry: Mathematical Foundations and Applications”, John Wiley & Sons, 1990. Zbl0689.28003 MR1102677 · Zbl 0689.28003
[14] C. E. Gutierrez - A. Montanari, Maximum and comparison principles for convex functions on the Heisenberg group, preprint. Zbl1056.35033 MR2103838 · Zbl 1056.35033 · doi:10.1081/PDE-200037752
[15] C. E. Gutierrez - A. Montanari, On the second order derivatives of convex functions on the Heisenberg group, preprint. MR2075987
[16] P. Lindquist - J. Manfredi - E. Saksman, Superharmonicity of nonlinear ground states, Rev. Math. Iberoam. 16 (2000), 17-27. Zbl0965.31002 MR1768532 · Zbl 0965.31002 · doi:10.4171/RMI/269 · eudml:39587
[17] G. Lu - J. Manfredi - B. Stroffolini, Convex functions on the Heisenberg group, to appear in Calc. Var. Partial Differential Equations. Zbl1072.49019 MR2027845 · Zbl 1072.49019 · doi:10.1007/s00526-003-0190-4
[18] V. Magnani, Lipschitz continuity, Aleksandrov theorem and characterizations for H-convex functions, preprint. Zbl1115.49004 MR2208954 · Zbl 1115.49004 · doi:10.1007/s00208-005-0717-4
[19] P. Mattila, “Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability”, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 1995. Zbl0819.28004 MR1333890 · Zbl 0819.28004
[20] J. Manfredi - B. Stroffolini, A version of the Hopf-Lax formula in the Heisenberg group, Comm. Partial Differential Equations 27 (2002), 1139-1159. Zbl1080.49023 MR1916559 · Zbl 1080.49023 · doi:10.1081/PDE-120004897
[21] P. Pansu, Métriques de Carnot-Carathéodory et quasiisométries des espaces symmétriques de rang un, Ann. Math. 129 (1989), 1-60. Zbl0678.53042 MR979599 · Zbl 0678.53042 · doi:10.2307/1971484
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