Sharp remainder of the Poincaré inequality
HTML articles powered by AMS MathViewer
- by Tohru Ozawa and Durvudkhan Suragan PDF
- Proc. Amer. Math. Soc. 148 (2020), 4235-4239 Request permission
Abstract:
In this paper, we obtain a sharp remainder formula for the Poincaré inequality which implies a simple proof of the sharp Poincaré inequality without using the variational principle. We also extend the idea to general Carnot groups. Thus, we have succeeded in finding a simple proof of the sharp Poincaré inequality in a more general case.References
- Amine Aribi and Sorin Dragomir, Dirichlet and Neumann eigenvalue problems on CR manifolds, Ric. Mat. 67 (2018), no. 2, 285–320. MR 3864776, DOI 10.1007/s11587-018-0420-x
- A. Bonfiglioli, E. Lanconelli, and F. Uguzzoni, Stratified Lie groups and potential theory for their sub-Laplacians, Springer Monographs in Mathematics, Springer, Berlin, 2007. MR 2363343
- Luca Capogna, Donatella Danielli, and Nicola Garofalo, An embedding theorem and the Harnack inequality for nonlinear subelliptic equations, Comm. Partial Differential Equations 18 (1993), no. 9-10, 1765–1794. MR 1239930, DOI 10.1080/03605309308820992
- Jingbo Dou, Pengcheng Niu, and Zixia Yuan, A Hardy inequality with remainder terms in the Heisenberg group and the weighted eigenvalue problem, J. Inequal. Appl. , posted on (2007), Art. ID 32585, 24. MR 2366343, DOI 10.1155/2007/32585
- Fausto Ferrari and Enrico Valdinoci, Some weighted Poincaré inequalities, Indiana Univ. Math. J. 58 (2009), no. 4, 1619–1637. MR 2542975, DOI 10.1512/iumj.2009.58.3601
- Tohru Ozawa and Hironobu Sasaki, Inequalities associated with dilations, Commun. Contemp. Math. 11 (2009), no. 2, 265–277. MR 2518581, DOI 10.1142/S0219199709003351
- Michael Ruzhansky and Durvudkhan Suragan, On horizontal Hardy, Rellich, Caffarelli-Kohn-Nirenberg and $p$-sub-Laplacian inequalities on stratified groups, J. Differential Equations 262 (2017), no. 3, 1799–1821. MR 3582213, DOI 10.1016/j.jde.2016.10.028
- Michael Ruzhansky and Durvudkhan Suragan, Hardy inequalities on homogeneous groups, Progress in Mathematics, vol. 327, Birkhäuser/Springer, Cham, 2019. 100 years of Hardy inequalities. MR 3966452, DOI 10.1007/978-3-030-02895-4
Additional Information
- Tohru Ozawa
- Affiliation: Department of Applied Physics, Waseda University, Tokyo 169-8555, Japan
- MR Author ID: 242556
- Email: txozawa@waseda.jp
- Durvudkhan Suragan
- Affiliation: Department of Mathematics, Nazarbayev University, 53 Kabanbay Batyr Avenue, Astana 010000, Kazakhstan
- MR Author ID: 864727
- Email: durvudkhan.suragan@nu.edu.kz
- Received by editor(s): October 24, 2019
- Published electronically: July 20, 2020
- Additional Notes: The authors were supported in parts by the Nazarbayev University program 091019CRP2120 and the Nazarbayev University grant 240919FD3901. No new data was collected or generated during the course of this research.
- Communicated by: Ariel Barton
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 148 (2020), 4235-4239
- MSC (2010): Primary 39B62, 39B99, 22E30
- DOI: https://doi.org/10.1090/proc/15119
- MathSciNet review: 4135292