Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

H-convex distributions in stratified groups


Authors: A. Bonfiglioli, E. Lanconelli, V. Magnani and M. Scienza
Journal: Proc. Amer. Math. Soc. 141 (2013), 3633-3638
MSC (2010): Primary 46F10; Secondary 26A51, 22E25
DOI: https://doi.org/10.1090/S0002-9939-2013-11653-9
Published electronically: July 15, 2013
MathSciNet review: 3080185
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We extend Dudley's characterization of convex functions to the framework of h-convex distributions on stratified groups. Precisely, we prove that every distribution with nonnegative horizontal Hessian is defined by an h-convex function.


References [Enhancements On Off] (What's this?)

  • 1. I.YA. BAKEL'MAN, Geometric methods of solution of elliptic equations (in Russian), Nauka, Moscow, (1965). MR 0194727 (33:2933)
  • 2. A. BONFIGLIOLI, E. LANCONELLI, Subharmonic functions on Carnot groups, Math. Ann. 325, 97-122, (2003). MR 1957266 (2004b:35047)
  • 3. A. BONFIGLIOLI, E. LANCONELLI, F. UGUZZONI, Stratified Lie groups and potential theory for their sub-Laplacians, Springer-Verlag, (2007). MR 2363343 (2009m:22012)
  • 4. D. DANIELLI, N. GAROFALO, D.M. NHIEU, Notions of convexity in Carnot groups, Comm. Anal. Geom. 11, n. 2, 263-341, (2003). MR 2014879 (2004m:22014)
  • 5. R.M. DUDLEY, On second derivatives of convex functions, Math. Scand. 41, 159-174, (1977). MR 0482164 (58:2250)
  • 6. G.B. FOLLAND, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat., 13, 161-207, (1975). MR 0494315 (58:13215)
  • 7. G.B. FOLLAND, E.M. STEIN, Hardy Spaces on Homogeneous Groups, Princeton University Press, (1982). MR 657581 (84h:43027)
  • 8. L. HSORMANDER, Hypoelliptic second order differential equations, Acta Math., 119, 147-171, (1967). MR 0222474 (36:5526)
  • 9. G. LU, J. MANFREDI, B. STROFFOLINI, Convex functions on the Heisenberg group, Calc. Var. Partial Differential Equations 19, no. 1, 1-22, (2004). MR 2027845 (2004m:35088)
  • 10. P. JUUTINEN, G. LU, J. J. MANFREDI, B. STROFFOLINI, Convex functions on Carnot groups, Rev. Mat. Iberoam. 23, no. 1, 191-200, (2007). MR 2351130 (2008m:49165)
  • 11. V. MAGNANI, Elements of Geometric Measure Theory on Sub-Riemannian Groups, Scuola Normale Superiore Pisa, (2002). MR 2115223 (2005m:28002)
  • 12. V. MAGNANI, Lipschitz continuity, Aleksandrov theorem and characterizations for h-convex functions, Math. Ann. 334, no. 1, 199-233, (2006). MR 2208954 (2007c:49005)
  • 13. V. MAGNANI, Towards differential calculus in stratified groups, to appear in J. Aust. Math. Soc.
  • 14. YU.G. RESHETNYAK, Generalized derivatives and differentiability almost everywhere, Mat. Sb. 75, 323-334 (in Russian), Math. USSR-Sb. 4, 293-302 (English translation), (1968). MR 0225159 (37:754)
  • 15. M. RICKLY, First-order regularity of convex functions on Carnot groups, J. Geom. Anal. 16, no. 4, 679-702, (2006). MR 2271949 (2008a:43007)
  • 16. L. SCHWARTZ, Théorie des distributions, Hermann, Paris, (1966). MR 0209834 (35:730)
  • 17. M. SCIENZA, Differentiability properties and characterizations of h-convex functions, Ph.D. Thesis, in progress.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 46F10, 26A51, 22E25

Retrieve articles in all journals with MSC (2010): 46F10, 26A51, 22E25


Additional Information

A. Bonfiglioli
Affiliation: Dipartimento Matematica, Università degli Studi di Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy
Email: bonfigli@dm.unibo.it

E. Lanconelli
Affiliation: Dipartimento Matematica, Università degli Studi di Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy
Email: lanconel@dm.unibo.it

V. Magnani
Affiliation: Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127, Pisa, Italy
Email: magnani@dm.unipi.it

M. Scienza
Affiliation: Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127, Pisa, Italy
Email: scienza@mail.dm.unipi.it

DOI: https://doi.org/10.1090/S0002-9939-2013-11653-9
Keywords: Distribution, convexity, stratified group
Received by editor(s): August 5, 2011
Received by editor(s) in revised form: January 11, 2012
Published electronically: July 15, 2013
Additional Notes: The third author was supported by the European Project ERC AdG *GeMeThNES*.
Communicated by: Tatiana Toro
Article copyright: © Copyright 2013 American Mathematical Society

American Mathematical Society