×

zbMATH — the first resource for mathematics

A new characterization of convexity in free Carnot groups. (English) Zbl 1272.31009
Summary: A characterization of convex functions in \(\mathbb R^N\) states that an upper semicontinuous function \(u\) is convex if and only if \(u(Ax)\) is subharmonic (with respect to the usual Laplace operator) for every symmetric positive definite matrix \(A\). The aim of this paper is to prove that an analogue of this result holds for free Carnot groups \(\mathbb G\) when considering convexity in the viscosity sense. In the subelliptic context of Carnot groups, the linear maps \(x\mapsto Ax\) of the Euclidean case must be replaced by suitable group isomorphisms \(x\mapsto T_A(x)\), whose differential preserves the first layer of the stratification of \(\operatorname{Lie}(\mathbb G)\).

MSC:
31C05 Harmonic, subharmonic, superharmonic functions on other spaces
26B25 Convexity of real functions of several variables, generalizations
43A80 Analysis on other specific Lie groups
35J70 Degenerate elliptic equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Zoltán M. Balogh and Matthieu Rickly, Regularity of convex functions on Heisenberg groups, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 2 (2003), no. 4, 847 – 868. · Zbl 1121.43007
[2] Andrea Bonfiglioli, Lifting of convex functions on Carnot groups and lack of convexity for a gauge function, Arch. Math. (Basel) 93 (2009), no. 3, 277 – 286. · Zbl 1181.22013 · doi:10.1007/s00013-009-0033-4 · doi.org
[3] Andrea Bonfiglioli and Ermanno Lanconelli, Subharmonic functions on Carnot groups, Math. Ann. 325 (2003), no. 1, 97 – 122. · Zbl 1017.31003 · doi:10.1007/s00208-002-0371-z · doi.org
[4] A. Bonfiglioli, E. Lanconelli, and F. Uguzzoni, Uniform Gaussian estimates for the fundamental solutions for heat operators on Carnot groups, Adv. Differential Equations 7 (2002), no. 10, 1153 – 1192. · Zbl 1036.35061
[5] A. Bonfiglioli, E. Lanconelli, and F. Uguzzoni, Stratified Lie groups and potential theory for their sub-Laplacians, Springer Monographs in Mathematics, Springer, Berlin, 2007. · Zbl 1128.43001
[6] Andrea Bonfiglioli and Francesco Uguzzoni, Families of diffeomorphic sub-Laplacians and free Carnot groups, Forum Math. 16 (2004), no. 3, 403 – 415. · Zbl 1065.35102 · doi:10.1515/form.2004.018 · doi.org
[7] Luca Capogna and Diego Maldonado, A note on the engulfing property and the \Gamma ^1+\?-regularity of convex functions in Carnot groups, Proc. Amer. Math. Soc. 134 (2006), no. 11, 3191 – 3199. · Zbl 1109.35028
[8] Luca Capogna, Scott D. Pauls, and Jeremy T. Tyson, Convexity and horizontal second fundamental forms for hypersurfaces in Carnot groups, Trans. Amer. Math. Soc. 362 (2010), no. 8, 4045 – 4062. · Zbl 1203.43007
[9] Dah-Yan Hwang, Some interpolations and refinements of Hadamard’s inequality for \?-convex functions in Carnot groups, Math. Inequal. Appl. 10 (2007), no. 2, 287 – 297. · Zbl 1180.26014 · doi:10.7153/mia-10-25 · doi.org
[10] Donatella Danielli, Nicola Garofalo, and Duy-Minh Nhieu, Notions of convexity in Carnot groups, Comm. Anal. Geom. 11 (2003), no. 2, 263 – 341. · Zbl 1077.22007 · doi:10.4310/CAG.2003.v11.n2.a5 · doi.org
[11] D. Danielli, N. Garofalo, D. M. Nhieu, and F. Tournier, The theorem of Busemann-Feller-Alexandrov in Carnot groups, Comm. Anal. Geom. 12 (2004), no. 4, 853 – 886. · Zbl 1071.22004
[12] G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (1975), no. 2, 161 – 207. · Zbl 0312.35026 · doi:10.1007/BF02386204 · doi.org
[13] Léonard Gallardo, Capacités, mouvement brownien et problème de l’épine de Lebesgue sur les groupes de Lie nilpotents, Probability measures on groups (Oberwolfach, 1981) Lecture Notes in Math., vol. 928, Springer, Berlin-New York, 1982, pp. 96 – 120 (French, with English summary). · Zbl 0483.60072
[14] Nicola Garofalo, Geometric second derivative estimates in Carnot groups and convexity, Manuscripta Math. 126 (2008), no. 3, 353 – 373. · Zbl 1387.35073 · doi:10.1007/s00229-008-0182-y · doi.org
[15] Nicola Garofalo and Federico Tournier, New properties of convex functions in the Heisenberg group, Trans. Amer. Math. Soc. 358 (2006), no. 5, 2011 – 2055. · Zbl 1102.35033
[16] Matthew Grayson and Robert Grossman, Models for free nilpotent Lie algebras, J. Algebra 135 (1990), no. 1, 177 – 191. · Zbl 0717.17006 · doi:10.1016/0021-8693(90)90156-I · doi.org
[17] Cristian E. Gutiérrez and Annamaria Montanari, Maximum and comparison principles for convex functions on the Heisenberg group, Comm. Partial Differential Equations 29 (2004), no. 9-10, 1305 – 1334. · Zbl 1056.35033 · doi:10.1081/PDE-200037752 · doi.org
[18] Cristian E. Gutiérrez and Annamaria Montanari, On the second order derivatives of convex functions on the Heisenberg group, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 3 (2004), no. 2, 349 – 366. · Zbl 1170.35352
[19] Marshall Hall Jr., A basis for free Lie rings and higher commutators in free groups, Proc. Amer. Math. Soc. 1 (1950), 575 – 581. · Zbl 0039.26302
[20] Lars Hörmander, Notions of convexity, Progress in Mathematics, vol. 127, Birkhäuser Boston, Inc., Boston, MA, 1994. · Zbl 0835.32001
[21] Petri Juutinen, Guozhen Lu, Juan J. Manfredi, and Bianca Stroffolini, Convex functions on Carnot groups, Rev. Mat. Iberoam. 23 (2007), no. 1, 191 – 200. · Zbl 1124.49024 · doi:10.4171/RMI/490 · doi.org
[22] Guozhen Lu, Juan J. Manfredi, and Bianca Stroffolini, Convex functions on the Heisenberg group, Calc. Var. Partial Differential Equations 19 (2004), no. 1, 1 – 22. · Zbl 1072.49019 · doi:10.1007/s00526-003-0190-4 · doi.org
[23] Valentino Magnani, Lipschitz continuity, Aleksandrov theorem and characterizations for \?-convex functions, Math. Ann. 334 (2006), no. 1, 199 – 233. · Zbl 1115.49004 · doi:10.1007/s00208-005-0717-4 · doi.org
[24] Roberto Monti and Matthieu Rickly, Geodetically convex sets in the Heisenberg group, J. Convex Anal. 12 (2005), no. 1, 187 – 196. · Zbl 1077.53030
[25] Matthieu Rickly, First-order regularity of convex functions on Carnot groups, J. Geom. Anal. 16 (2006), no. 4, 679 – 702. · Zbl 1103.43005 · doi:10.1007/BF02922136 · doi.org
[26] Mingbao Sun and Xiaoping Yang, Generalized Hadamard’s inequality and \?-convex functions in Carnot groups, J. Math. Anal. Appl. 294 (2004), no. 2, 387 – 398. · Zbl 1069.26021 · doi:10.1016/j.jmaa.2003.10.050 · doi.org
[27] Ming-bao Sun and Xiao-ping Yang, Inequalities of Hadamard type for \?-convex functions in Carnot groups, Acta Math. Appl. Sin. Engl. Ser. 20 (2004), no. 1, 123 – 132. · Zbl 1135.22300 · doi:10.1007/s10255-004-0155-1 · doi.org
[28] Mingbao Sun and Xiaoping Yang, Lipschitz continuity for \?-convex functions in Carnot groups, Commun. Contemp. Math. 8 (2006), no. 1, 1 – 8. · Zbl 1119.43006 · doi:10.1142/S0219199706002015 · doi.org
[29] Mingbao Sun and Xiaoping Yang, Quasi-convex functions in Carnot groups, Chin. Ann. Math. Ser. B 28 (2007), no. 2, 235 – 242. · Zbl 1123.43004 · doi:10.1007/s11401-005-0052-9 · doi.org
[30] V. S. Varadarajan, Lie groups, Lie algebras, and their representations, Graduate Texts in Mathematics, vol. 102, Springer-Verlag, New York, 1984. Reprint of the 1974 edition. · Zbl 0955.22500
[31] N. Th. Varopoulos, L. Saloff-Coste, and T. Coulhon, Analysis and geometry on groups, Cambridge Tracts in Mathematics, vol. 100, Cambridge University Press, Cambridge, 1992. · Zbl 0813.22003
[32] Changyou Wang, Viscosity convex functions on Carnot groups, Proc. Amer. Math. Soc. 133 (2005), no. 4, 1247 – 1253. · Zbl 1057.22012
[33] Frank W. Warner, Foundations of differentiable manifolds and Lie groups, Graduate Texts in Mathematics, vol. 94, Springer-Verlag, New York-Berlin, 1983. Corrected reprint of the 1971 edition. · Zbl 0516.58001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.