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Quasi-convex functions in Carnot groups. (English) Zbl 1123.43004
The authors introduce the concept of \(h\)-quasiconvexity which generalizes the notion of \(h\)-convexity in the Carnot group \(G\). An example of \(h\)-quasiconvex function which is not \(h\)-convex is provided. Some interesting properties similar to those of \(h\)-convex functions on \(G\) are given. In particular, the authors show that the notions of \(h\)-quasiconvex functions and \(h\)-convex sets are equivalent, give the \(L^\infty\) estimates of the first derivatives of \(h\)-quasiconvex functions, and prove that \(h\)-quasiconvex functions on a Carnot group \(G\) of step two are locally bounded from above. Furthermore, for a Carnot group \(G\) of step two, the authors obtain that \(h\)-convex functions are locally Lipschitz continuous and twice differentiable almost everywhere in \(G\). This last result is the version of the Busemann-Feller-Alexandrov theorem for the class of \(h\)-convex functions in Carnot groups of step two.

MSC:
43A80 Analysis on other specific Lie groups
43-06 Proceedings, conferences, collections, etc. pertaining to abstract harmonic analysis
26B25 Convexity of real functions of several variables, generalizations
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[1] Arrow, K. J. and Enthoven, A. C., Quasi-concave programming, Economitrica, 29, 1961, 779–800 · Zbl 0104.14302 · doi:10.2307/1911819
[2] Balogh, Z. M. and Rickly, M., Regularity of convex functions on Heisenberg groups, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 2(5), 2003, 847–868 · Zbl 1121.43007
[3] Bellaïche, A. and Risler, J.-J., Sub-Riemannian Geometry, Progress in Mathematics, Vol. 144, Birkhauser, 1996
[4] Cabre, X. and Caffarelli, L., Fully nonlinear elliptic equations, AMS Colloquium Publications, 43, AMS, Providence, RI, 1995
[5] Crandall, M., Ishii, H. and Lions, P. L., User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27(1), 1992, 1–67 · Zbl 0755.35015 · doi:10.1090/S0273-0979-1992-00266-5
[6] Danielli, D., Garofalo, N. and Nhieu, D. M., Notions of convexity in Carnot groups, Comm. Analysis and Geometry, 11(2), 2003, 263–341 · Zbl 1077.22007
[7] Danielli, D., Garofalo, N., Nhieu, D. M. and Tournier, F., The theorem of Busemann-Feller-Alexandrov in Carnot groups, Comm. Analysis and Geometry, 12(4), 2004, 853–886 · Zbl 1071.22004
[8] Garofalo, N. and Nhieu, D. M., Lipschitz continuity, global smooth approximations and extension theorems for Sobolev functions in Carnot-Carathèodory spaces, J. Anal. Math., 74, 1998, 67–97 · Zbl 0906.46026 · doi:10.1007/BF02819446
[9] Fenchel, W., Convex Cones, Sets and Functions, Princeton University, Princeton, New Jersey, 1951 · Zbl 0053.12203
[10] Ferland, J. A., Matrix-theoretic criteria for the quasi-convexity of twice continuously differenciable functions, Linear Alg. Appl., 38, 1981, 51–63 · Zbl 0468.15008 · doi:10.1016/0024-3795(81)90007-0
[11] Folland, G. B., Subelliptic estimates and function space on nilpotent Lie groups, Ark. Math., 13, 1975, 161–207 · Zbl 0312.35026 · doi:10.1007/BF02386204
[12] Folland, G. B. and Stein, E. M., Hardy Space on Homogeneous Groups, Princeton University Press, Princeton, New Jersey, 1982 · Zbl 0508.42025
[13] Greenberg, H. J. and Pierskalla, W. P., A review of quasi-convex functions, Operation Research, 19, 1971, 1553–1570 · Zbl 0228.26012 · doi:10.1287/opre.19.7.1553
[14] Lu, G., Manfredi, J. and Stroffolini, B., Convex functions on the Heisenberg group, Calc. Var. Partial Differential Equations, 19, 2003, 1–22 · Zbl 1072.49019 · doi:10.1007/s00526-003-0190-4
[15] Magnani, V., Lipschitz continuity, Alexandrov theorem, and characterizations for H-convex functions, Math. Annalen., 334(1), 2006, 199–233 · Zbl 1115.49004 · doi:10.1007/s00208-005-0717-4
[16] Nikaido, H., On Von Neumann’s minimax theorem, Pacific J. Math., 4, 1954, 65–72 · Zbl 0055.10004
[17] Pansu, P., Métriques de Carnot-Carathéodory et quasii-sométries des espacec symétriques de rang un, Ann. Math., 129, 1989, 1–60 · Zbl 0678.53042 · doi:10.2307/1971484
[18] Stein, E. M., Harmonic Analysis: Real VaribleMethods, Orthogonality and Oscillatory Integrals, Princeton University Press, Princeton, 1993
[19] Sun, M. and Yang, X., Inequalities of Hadamard type for r-convex functions in Carnot groups, Acta Math. Appl. Sin., 20(1), 2004, 123–132 · Zbl 1135.22300 · doi:10.1007/s10255-004-0155-1
[20] Sun, M. and Yang, X., Some properties of quasiconvex functions on the Heisenberg groups, Acta Math. Appl. Sin., 21(4), 2005, 571–580 · Zbl 1102.43003 · doi:10.1007/s10255-005-0266-3
[21] Sun, M. and Yang, X., Lipschitz continuity for H-Convex functions in Carnot groups, Commun. Contem- porary Mathematics, 8(1), 2006, 1–8 · Zbl 1119.43006 · doi:10.1142/S0219199706002015
[22] Varadarajan, V. S., Lie Groups, Lie Algebras, and Their Representions, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1974 · Zbl 0371.22001
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