A note on the engulfing property and the Γ^{1+𝛼}-regularity of convex functions in Carnot groups

Capogna, Luca; Maldonado, Diego

doi:10.1090/S0002-9939-06-08359-6

A note on the engulfing property and the $\Gamma ^{1+ \alpha }$-regularity of convex functions in Carnot groups
HTML articles powered by AMS MathViewer

by Luca Capogna and Diego Maldonado PDF

Proc. Amer. Math. Soc. 134 (2006), 3191-3199 Request permission

Abstract:

We study the engulfing property for convex functions in Carnot groups. As an application we show that the horizontal gradient of functions with this property is Hölder continuous.

References

Luis A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math. (2) 130 (1989), no. 1, 189–213. MR 1005611, DOI 10.2307/1971480
Luis A. Caffarelli, Interior $W^{2,p}$ estimates for solutions of the Monge-Ampère equation, Ann. of Math. (2) 131 (1990), no. 1, 135–150. MR 1038360, DOI 10.2307/1971510
Luis A. Caffarelli, Some regularity properties of solutions of Monge Ampère equation, Comm. Pure Appl. Math. 44 (1991), no. 8-9, 965–969. MR 1127042, DOI 10.1002/cpa.3160440809
Donatella Danielli, Nicola Garofalo, and Duy-Minh Nhieu, Notions of convexity in Carnot groups, Comm. Anal. Geom. 11 (2003), no. 2, 263–341. MR 2014879, DOI 10.4310/CAG.2003.v11.n2.a5
Donatella Danielli, Nicola Garofalo, and Duy-Minh Nhieu, On the best possible character of the $L^Q$ norm in some a priori estimates for non-divergence form equations in Carnot groups, Proc. Amer. Math. Soc. 131 (2003), no. 11, 3487–3498. MR 1991760, DOI 10.1090/S0002-9939-03-07105-3
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. MR 2104079
G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (1975), no. 2, 161–207. MR 494315, DOI 10.1007/BF02386204
G. B. Folland and E. M. Stein, Estimates for the $\bar \partial _{b}$ complex and analysis on the Heisenberg group, Comm. Pure Appl. Math. 27 (1974), 429–522. MR 367477, DOI 10.1002/cpa.3160270403
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. MR 657581
L. Forzani & D. Maldonado, Doubling measures, quasi-symmetric mappings, and a class of convex functions on the real line, submitted.
Liliana Forzani and Diego Maldonado, On geometric characterizations for Monge-Ampère doubling measures, J. Math. Anal. Appl. 275 (2002), no. 2, 721–732. MR 1943775, DOI 10.1016/S0022-247X(02)00389-X
Liliana Forzani and Diego Maldonado, Properties of the solutions to the Monge-Ampère equation, Nonlinear Anal. 57 (2004), no. 5-6, 815–829. MR 2067735, DOI 10.1016/j.na.2004.03.019
Nicola Garofalo and Duy-Minh Nhieu, Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces, Comm. Pure Appl. Math. 49 (1996), no. 10, 1081–1144. MR 1404326, DOI 10.1002/(SICI)1097-0312(199610)49:10<1081::AID-CPA3>3.0.CO;2-A
N. Garofalo & F. Tournier, Subelliptic estimates for fully nonlinear equations in the Heisenberg group, to appear in Trans. Amer. Math. Soc.
Cristian E. Gutiérrez, The Monge-Ampère equation, Progress in Nonlinear Differential Equations and their Applications, vol. 44, Birkhäuser Boston, Inc., Boston, MA, 2001. MR 1829162, DOI 10.1007/978-1-4612-0195-3
Cristian E. Gutiérrez and Qingbo Huang, Geometric properties of the sections of solutions to the Monge-Ampère equation, Trans. Amer. Math. Soc. 352 (2000), no. 9, 4381–4396. MR 1665332, DOI 10.1090/S0002-9947-00-02491-0
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. MR 2075987
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. MR 2103838, DOI 10.1081/PDE-200037752
Lars Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147–171. MR 222474, DOI 10.1007/BF02392081
Steven G. Krantz, Structure and interpolation theorems for certain Lipschitz spaces and estimates for the $\overline \partial$ equation, Duke Math. J. 43 (1976), no. 2, 417–439. MR 430311
Steven G. Krantz, Geometric Lipschitz spaces and applications to complex function theory and nilpotent groups, J. Functional Analysis 34 (1979), no. 3, 456–471. MR 556266, DOI 10.1016/0022-1236(79)90087-9
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. MR 2027845, DOI 10.1007/s00526-003-0190-4
Alexander Nagel, Elias M. Stein, and Stephen Wainger, Balls and metrics defined by vector fields. I. Basic properties, Acta Math. 155 (1985), no. 1-2, 103–147. MR 793239, DOI 10.1007/BF02392539
Pierre Pansu, Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. of Math. (2) 129 (1989), no. 1, 1–60 (French, with English summary). MR 979599, DOI 10.2307/1971484
Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192