×

zbMATH — the first resource for mathematics

Lipschitz continuity, Aleksandrov theorem and characterizations for \(H\)-convex functions. (English) Zbl 1115.49004
The author proves that in stratified groups \(H\)-convex functions locally bounded from above are locally Lipschitz; moreover, he proves that the class of \(v\)-convex functions corresponds to the class of \(H\)-convex functions that are upper semicontinuous, and then they are also Lipschitz. In the class of step 2 groups a more precise result is given; the characterization of locally Lipschitz \(H\)-convex functions as measures whose distributional horizontal Hessian is positive semidefinite.
These results are the generalization of the Euclidean results given by R. M. Dudley [Math. Scand. 41, no. 1, 159–174 (1977; Zbl 0386.46037)] and Yu. G. Reshetnyak [Mat. Sb. (N.S.) 75 (117), 323–334 (1968; Zbl 0176.12001)].

MSC:
49J27 Existence theories for problems in abstract spaces
26A51 Convexity of real functions in one variable, generalizations
26B25 Convexity of real functions of several variables, generalizations
52A41 Convex functions and convex programs in convex geometry
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Ambrosio, L., Magnani, V.: Weak differentiability of BV functions on stratified groups. Math. Z. 245, 123–153 (2003) · Zbl 1048.49030 · doi:10.1007/s00209-003-0530-2
[2] Ambrosio, L., Fusco, N., Pallara, D.: Functions of bounded variation and free discontinuity problems. Oxford University Press, 2000 · Zbl 0957.49001
[3] Bakel’man, I.Ya.: Geometric methods of solutions of elliptic equations, (in Russian). Nauka, Moscow, 1965
[4] Balogh, Z., Rickly, M.: Regularity of convex functions on Heisenberg groups, preprint (2003) · Zbl 1121.43007
[5] Bellaïche, A., Risler, J.J. eds,: Sub-Riemannian geometry, Progress in Mathematics, 144, Birkhäuser Verlag, Basel, 1996 · Zbl 0848.00020
[6] Bianchi, G., Colesanti, A., Pucci, C.: On the second order differentiability of convex surfaces. Geom. Dedicata 60, 39–48 (1996) · Zbl 0843.26007 · doi:10.1007/BF00150866
[7] Bieske, T.: On harmonic functions on the Heisenberg group. Comm. Partial Differential Equations 27(3,4), 727–762 (2002) · Zbl 1090.35063 · doi:10.1081/PDE-120002872
[8] Caffarelli, L.A., Cabré, X.: Fully nonlinear elliptic equations. AMS Colloquium Publications 43, AMS, Providence, RI, 1995 · Zbl 0834.35002
[9] Danielli, D., Garofalo, N., Nhieu, D.M.: Notions of convexity in Carnot groups. Comm. Anal. Geom. 11(2), 263–341 (2003) · Zbl 1077.22007
[10] Danielli, D., Garofalo, N., Nhieu, D.M., Tournier, F.: The theorem of Busemann- Feller-Alexandrov in Carnot groups. Comm. Anal. Geom. 12(4), 853–886 (2004) · Zbl 1071.22004
[11] Dudley, R.M.: On second derivatives of convex functions. Math. Scand. 41(1), 159–174 (1977) · Zbl 0386.46037
[12] Evans, L.C., Gariepy, R.F.: Measure theory and fine properties of functions. Studies in Advanced Mathematics, CRC Press, Boca Raton, 1992 · Zbl 0804.28001
[13] Federer, H.: Geometric Measure Theory. Springer, 1969 · Zbl 0176.00801
[14] Folland, G.B., Stein, E.M.: Hardy Spaces on Homogeneous groups. Princeton University Press, 1982 · Zbl 0508.42025
[15] Garofalo, N., Nhieu, D.M.: Lipschitz continuity, global smooth approximation and extension theorems for Sobolev functions in Carnot-Carathéodory spaces. Jour. Anal. Math. 74, 67–97 (1998) · Zbl 0906.46026 · doi:10.1007/BF02819446
[16] Gutirrez, C.E., Montanari, A.: On the second order derivatives of convex functions on the Heisenberg group. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 3(2), 349–366 (2004) · Zbl 1170.35352
[17] Gromov, M.: Carnot-Carathéodory spaces seen from within. Subriemannian Geometry, Progress in Mathematics, 144 ed. by A. Bellaiche and J. Risler, Birkhauser Verlag, Basel, 1996 · Zbl 0864.53025
[18] Hajlasz, P., Koskela, P.: Sobolev met Poincaré. Mem. Amer. Math. Soc. 145, (2000)
[19] Hochschild, G.: The structure of Lie groups. Holden-Day, 1965 · Zbl 0131.02702
[20] Hörmander, L.: The Analysis of linear partial differential operators. Springer-Verlag, 1990 · Zbl 0687.35002
[21] Juutinen, P., Lu, G., Manfredi, J., Stroffolini, B.: Convex functions in Carnot groups, preprint (2005) · Zbl 1124.49024
[22] Krylov, N.V.: Nonlinear elliptic and parabolic equations of the second order. Mathematics and its Applications, Reidel, 1987 · Zbl 0619.35004
[23] Lu, G., Manfredi, J., Stroffolini, B.: Convex functions on the Heisenberg group. Calc. Var. Partial Differential Equations 19(1), 1–22 (2004) · Zbl 1072.49019 · doi:10.1007/s00526-003-0190-4
[24] Lu, G., Manfredi, J., Stroffolini, B.: Convex functions in Carnot groups, forthcoming · Zbl 1124.49024
[25] Magnani, V.: Differentiability from Sobolev-Poincaré inequality. Studia Math. 168(3), 251–272 (2005) · Zbl 1097.26013 · doi:10.4064/sm168-3-5
[26] Monti, R., Serra Cassano, F.: Surface measures in Carnot-Carathédory spaces. Calc. Var. 13(3), 339–376 (2001) · Zbl 1032.49045 · doi:10.1007/s005260000076
[27] Nagel, A., Stein, E.M., Wainger, S.: Balls and metrics defined by vector fields I: Basic properties. Acta Math. 155, 103–147 (1985) · Zbl 0578.32044 · doi:10.1007/BF02392539
[28] Pansu, P.: Métriques de Carnot-Carathéodory quasiisométries des espaces symétriques de rang un. Ann. Math. 129, 1–60 (1989) · Zbl 0678.53042 · doi:10.2307/1971484
[29] Reshetnyak, Yu.G.: Generalized derivatives and differentiability almost everywhere. Mat. Sb. 75, 323–334 (1968) (in Russian), Math. USSR-Sb. 4, 293–302 (English translation)
[30] Rickly, M.: On questions of existence and regularity related to notions of convexity in Carnot groups, Ph.D. thesis (2005)
[31] Rickly, M.: First order regularity of convex functions on Carnot groups, preprint (2005) · Zbl 1103.43005
[32] Stein, E.M.: Harmonic Analysis. Princeton Univ. Press, 1993 · Zbl 0821.42001
[33] Trudinger, N.: On Hessian measures for non-commuting vector fields, ArXiv preprint: math.AP/0503695, (2005)
[34] Varadarajan, V.S.: Lie groups, Lie algebras and their representation. Springer-Verlag, New York, 1984 · Zbl 0955.22500
[35] Varopoulos, N.Th., Saloff-Coste, L., Coulhon, T.: Analysis and Geometry on Groups. Cambridge University Press, Cambridge, 1992 · Zbl 0813.22003
[36] Wang, C.: Viscosity convex functions on Carnot groups, 133(4), 1247–1253 (2005) · Zbl 1057.22012
[37] Wang, C.: The Euler Equations of absolutely minimizing Lipschitz extensions for vector fields satisfying Hörmander condition, preprint (2005)
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.