Regularity of convex functions on Heisenberg groups

Balogh, Zoltán M.; Rickly, Matthieu

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 2 (2003) no. 4, pp. 847-868.

We discuss differentiability properties of convex functions on Heisenberg groups. We show that the notions of horizontal convexity (h-convexity) and viscosity convexity (v-convexity) are equivalent and that h-convex functions are locally Lipschitz continuous. Finally we exhibit Weierstrass-type h-convex functions which are nowhere differentiable in the vertical direction on a dense set or on a Cantor set of vertical lines.

[1] O. Alvarez - J.-M. Larsy - P.-L. Lions, Convex viscosity solutions and state constraints, J. Math. Pures Appl. (76) 9 (1997), 265-288. | MR | Zbl

[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. | MR | Zbl

[3] L. Ambrosio - V. Magnani, Weak differentiability of BV functions on stratified groups, Math. Z. (1) 245 (2003), 123-153. | MR | Zbl

[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. | MR | Zbl

[5] T. Bieske, On $\infty$-harmonic functions on the Heisenberg group, Comm. Partial Differential Equations 27 (2002), 727-762. | MR | Zbl

[6] X. Cabre - L. Caffarelli, Fully nonlinear elliptic equations, AMS colloquium publications 43, AMS, Providence, RI, 1995. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[9] J. Cygan, Subadditivity of homogeneous norms on certain nilpotent Lie groups, Proc. Amer. Math. Soc. 83 (1981), 69-70. | MR | Zbl

[10] D. Danielli - N. Garofalo - D.-M. Nhieu, Notions of convexity in Carnot groups, Comm. Anal. Geom. (2) 11 (2003), 263-341. | MR | Zbl

[11] D. Danielli - N. Garofalo - D.-M. Nhieu - F. Tournier, The theorem of Busemann-Feller-Alexandrov in Carnot groups, preprint. | MR | Zbl

[12] L. Evans - R. Gariepy, “Measure Theory and Fine Properties of Functions”, Studies in Advanced Mathematics, CRC Press, 1992. | MR | Zbl

[13] K. J. Falconer, “Fractal Geometry: Mathematical Foundations and Applications”, John Wiley & Sons, 1990. | MR | Zbl

[14] C. E. Gutierrez - A. Montanari, Maximum and comparison principles for convex functions on the Heisenberg group, preprint. | MR | Zbl

[15] C. E. Gutierrez - A. Montanari, On the second order derivatives of convex functions on the Heisenberg group, preprint. | Numdam | MR

[16] P. Lindquist - J. Manfredi - E. Saksman, Superharmonicity of nonlinear ground states, Rev. Math. Iberoam. 16 (2000), 17-27. | MR | Zbl

[17] G. Lu - J. Manfredi - B. Stroffolini, Convex functions on the Heisenberg group, to appear in Calc. Var. Partial Differential Equations. | MR | Zbl

[18] V. Magnani, Lipschitz continuity, Aleksandrov theorem and characterizations for H-convex functions, preprint. | MR | Zbl

[19] P. Mattila, “Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability”, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 1995. | MR | Zbl

[20] J. Manfredi - B. Stroffolini, A version of the Hopf-Lax formula in the Heisenberg group, Comm. Partial Differential Equations 27 (2002), 1139-1159. | MR | Zbl

[21] P. Pansu, Métriques de Carnot-Carathéodory et quasiisométries des espaces symmétriques de rang un, Ann. Math. 129 (1989), 1-60. | MR | Zbl