Share This Article:

On Riesz Mean Inequalities for Subelliptic Laplacian

Abstract Full-Text HTML Download Download as PDF (Size:87KB) PP. 694-698
DOI: 10.4236/am.2011.26091    4,406 Downloads   7,722 Views   Citations

ABSTRACT

In this paper, we mainly focus on the Riesz means of eigenvalues of the subelliptic Laplacian on the Heisenberg group Hn. We establish a trace formula of associated eigenvalues, then we prove differential inequalities, difference inequalities and monotonicity formulas for the Riesz means of eigenvalues of the subelliptic Laplacian.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

G. Jia, J. Wang and Y. Xiong, "On Riesz Mean Inequalities for Subelliptic Laplacian," Applied Mathematics, Vol. 2 No. 6, 2011, pp. 694-698. doi: 10.4236/am.2011.26091.

References

[1] G. N. Hile and M. H. Protter, “Inequalities for Eigenvalues of the Laplacian,” Indiana University Mathematics Journal, Vol. 29, No. 4, 1980, pp. 523-538. doi:10.1512/iumj.1980.29.29040
[2] E. M. Harrell II and J. Stubbe, “On Trace Identities and Universal Eigenvalue Estimates for Some Partial Differential Operators,” Transactions of the American Mathematical Society, Vol. 349, No. 5, 1997, pp. 1797-1809. doi:10.1090/S0002-9947-97-01846-1
[3] E. M. Harrell and L. Hermi, “Differential Inequalities for Riesz Means and Weyl-Type Bounds for Eigenvalues,” Journal of Functional Analysis, Vol. 254, No. 12, 2008, pp. 3173-3191. doi:10.1016/j.jfa.2008.02.016
[4] A. El Soufi, E. M. Harrell and S. Ilias, “Universal Inequalities for the Eigenvalues of Laplace and Schr?inger Operators on Submanifolds,” American Mathematical Society, Providence, Vol. 361, 2009, pp. 2337-2350.
[5] M. S. Ashbaugh and L. Hermi, “Universal Inequalities for Higher-Order Elliptic Operators,” Proceedings of the American Mathematical Society, Vol. 126, 1998 pp. 2623-2630. doi:10.1090/S0002-9939-98-04707-8
[6] P. Lévy-Bruhl, “Résolubilité Locale et Globale d’Opérateurs Invariants du Second Ordre sur des Groupes de Lie Nilpotents,” Bulletin des Sciences Mathematiques, Vol. 104, 1980, pp. 369-391.
[7] D. Müller and M. M. Peloso, “Non-Solvability for a Class of Left-Invariant Second-Order Differential Operators on the Heisenberg Group,” Transactions of the American Mathematical Society, Vol. 355, No. 5, 2003, pp. 2047- 2064. doi:10.1090/S0002-9947-02-03232-4
[8] P. C. Niu and H. Q. Zhang, “Payne-Polya-Weinberge Type Inequalities for Eigenvalues of Nonelliptic Operators,” Pacific Journal of Mathematics, Vol. 208, No. 2, 2003, pp. 325-345. doi:10.2140/pjm.2003.208.325
[9] G. Jia, “On Some Function Spaces and Variational Problems Related to the Heisenberg Group,” Ph.D. Thesis, 2003.
[10] M. S. Ashbaugh and L. Hermi, “A Unified Approach to Universal Inequalities for Eigenvalues of Elliptic Operators,” Pacific Journal of Mathematics, Vol. 217, No. 2, 2004, pp. 201-220. doi:10.2140/pjm.2004.217.201
[11] E. M. Harrell and L. Hermi, “On Riesz Means of Eigenvalues,” Eprint arXiv: 0712. 4088.
[12] B. Helffer and D. Robert, “Riesz Means of Bounded States and Semi-Classical Limit Connected with a Lieb-Thirring Conjecture, II, Ann. Inst. H. Poincar,” Phys. Théor., Vol. 53, 1990, pp. 139-147

  
comments powered by Disqus

Copyright © 2020 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.