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Viscosity convex functions on Carnot groups. (English) Zbl 1057.22012
A simply connected (nilpotent) Lie group $$G$$ with Lie algebra $$\mathfrak{g}$$ decomposed as $$\mathfrak{g} = V_1 + V_2 + \cdots + V_r$$ with $$[V_1, V_j] = V_{j+1}$$ for $$j < r$$ and $$[V_1,V_r] = 0$$ is called a Carnot group. Given a domain $$\Omega\subset G$$, a function $$u:\Omega\to\mathbb{R}$$ is said to be $$h$$-convex whenever $$u| [p, q]$$ is convex for any line segment $$[p, q] = \exp ([0, 1]\cdot X)\subset\Omega$$ with $$X\in V_1$$. If $$u:\Omega\to\mathbb{R}$$ is upper semicontinuous, then $$u$$ is said to be $$v$$-convex whenever its horizontal hessian $$\nabla ^2_h u = \nabla ^2u| V_1\times V_1$$ is nonnegative in the sense of viscosity; that is $$\sum _{i,j=1}^m \xi_i\xi_jX_iX_j\phi (p)\neq 0$$ for any $$p\in\Omega$$, $$\xi_1, \dots , \xi_m\in\mathbb{R}$$ ($$m = \dim V_1$$), a basis $$X_1, \dots X_m$$ of $$V_1$$, and any C$$^2$$-function $$\phi$$ “touching” $$u$$ at $$p$$ from above. Theorem. Any upper semicontinuous $$v$$-convex function on a Carnot group is $$h$$-convex.

##### MSC:
 2.2e+31 Analysis on real and complex Lie groups 2.2e+26 Nilpotent and solvable Lie groups
##### Keywords:
Carnot group; convex function
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##### References:
 [1] Thomas Bieske, On \infty -harmonic functions on the Heisenberg group, Comm. Partial Differential Equations 27 (2002), no. 3-4, 727 – 761. · Zbl 1090.35063 · doi:10.1081/PDE-120002872 · doi.org [2] 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 [3] Luis A. Caffarelli and Xavier Cabré, Fully nonlinear elliptic equations, American Mathematical Society Colloquium Publications, vol. 43, American Mathematical Society, Providence, RI, 1995. · Zbl 0834.35002 [4] Michael G. Crandall, Hitoshi Ishii, and Pierre-Louis Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 1 – 67. · Zbl 0755.35015 [5] 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 [6] Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. · Zbl 0804.28001 [7] G. B. Folland and Elias M. Stein, Hardy spaces on homogeneous groups, Mathematical Notes, vol. 28, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1982. · Zbl 0508.42025 [8] R. Jensen, P.-L. Lions, and P. E. Souganidis, A uniqueness result for viscosity solutions of second order fully nonlinear partial differential equations, Proc. Amer. Math. Soc. 102 (1988), no. 4, 975 – 978. · Zbl 0662.35048 [9] P. Juutinen, G. Lu, J. Manfredi, B. Stroffolini. In preparation. [10] 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 [11] C. Y. Wang, The Aronsson equation for absolute minimizers of $$L^{\infty}$$-functionals associated with vector fields satisfying Hörmander’s condition. To appear in Trans. Amer. Math. Soc. · Zbl 1192.35038
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