Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-15T19:23:01.542Z Has data issue: false hasContentIssue false
  • English
  • Français

Some Functional Inequalities on Polynomial Volume Growth Lie Groups

Published online by Cambridge University Press:  20 November 2018

Diego Chamorro
Diego Chamorro*
Affiliation:
Laboratoire d’Analyse et de Probabilités, Université d’Evry Val d’Essonne et ENSIIE, 91025 Evry Cedex email: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this article we study some Sobolev-type inequalities on polynomial volume growth Lie groups. We show in particular that improved Sobolev inequalities can be extended to this general framework without the use of the Littlewood–Paley decomposition.

Keywords

22E30Sobolev inequalitiespolynomial volume growth Lie groups
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 64 , Issue 3 , 01 June 2012 , pp. 481 - 496
Copyright
Copyright © Canadian Mathematical Society 2012

References

[1] Badr, N., Gagliardo-Nirenberg inequalities on manifolds. J. Math. Anal. Appl. 349(2009), no. 2, 493–502. http://dx.doi.org/10.1016/j.jmaa.2008.09.013 Google Scholar
[2] Bahouri, H., Gérard, P., and Xu, C-J, Espaces de Besov et estimations de Strichartz généralisées sur le groupe de Heisenberg. J. Anal. Math. 82(2000), 93–118. http://dx.doi.org/10.1007/BF02791223 Google Scholar
[3] Chamorro, D., Improved Sobolev inequalities and Muckenhoupt weights on stratified Lie groups. J. Math. Anal. Appl. 377(2011), no. 2, 695–709. http://dx.doi.org/10.1016/j.jmaa.2010.11.047 Google Scholar
[4] Cohen, A., Dahmen, W., Daubechies, I. and De Vore, R., Harmonic analysis of the space BV. Rev. Mat. Iberoamericana 19(2003), no. 1, 235–263.Google Scholar
[5] Cowling, M., Gaudry, G., Giulini, S., and Mauceri, G., Weak type (1, 1) estimates for heat kernel maximal funtions on Lie groups. Trans. Amer. Math. Soc. 323(1991), no. 2, 637–649. http://dx.doi.org/10.2307/2001548 Google Scholar
[6] Folland, G., Subelliptic estimates and function spaces on nilpotent Lie groups. Ark. Mat. 13(1975), no. 2, 161–208. http://dx.doi.org/10.1007/BF02386204 Google Scholar
[7] Folland, G. and Stein, E. M., Hardy Spaces on Homogeneous Groups. Mathematical Notes 28. Princeton University Press, Princeton, NJ, 1982.Google Scholar
[8] Furioli, G., Melzi, C., and Veneruso, A., Littlewood-Paley decomposition and Besov spaces on Lie groups of polynomial growth. Math. Nachr. 279(2006), no. 9-10, 1028–1040. http://dx.doi.org/10.1002/mana.200510409 Google Scholar
[9] Gérard, P., Meyer, Y., and Oru, F., Inégalités de Sobolev Précisées. In: Séminaire sur les Équations aux Dérivées Partielles, 1996-1997. École Polytech., Palaiseau, 1997.Google Scholar
[10] Hulanicki, A., A functional calculus for Rockland operators on nilpotent Lie groups. Studia Math. 78 (1984), no. 3, 253–266.Google Scholar
[11] Ledoux, M., On improved Sobolev embedding theorems. Math. Res. Lett. 10(2003), no. 5-6, 659–669.Google Scholar
[12] Saka, K., Besov Spaces and Sobolev spaces on a nilpotent Lie group. Tôhoku. Math. J. 31(1979), no. 4, 383–437. http://dx.doi.org/10.2748/tmj/1178229728 Google Scholar
[13] Stein, E. M., Topics in Harmonic Analysis. Annals of Mathematics Studies 63. Princeton University Press, Princeton, NJ, 1970.Google Scholar
[14] Stein, E. M., Harmonic Analysis. Princeton Mathematical Series 43. Princeton University Press, Princeton, NJ, 1993.Google Scholar
[15] Varopoulos, N. T., Saloff-Coste, L., and Coulhon, T., Analysis and Geometry on Groups. Cambridge Tracts in Mathematics 100. Cambridge University Press, Cambridge, 1992.Google Scholar