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Riesz Means of Dirichlet Eigenvalues for the Sub-Laplace Operator on the Engel Group

DOI: 10.4236/apm.2013.39A2001    3,559 Downloads   5,712 Views  
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ABSTRACT

In this paper, we are concerned with the Riesz means of Dirichlet eigenvalues for the sub-Laplace operator on the Engel group and deriver different inequalities for Riesz means. The Weyl-type estimates for means of eigenvalues are given.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

J. Xue, "Riesz Means of Dirichlet Eigenvalues for the Sub-Laplace Operator on the Engel Group," Advances in Pure Mathematics, Vol. 3 No. 9B, 2013, pp. 1-7. doi: 10.4236/apm.2013.39A2001.

References

[1] N. Garofalo and F. Tournier, “New Properties of Convex Functions in the Heisenberg Group,” Transactions of the American Mathematical Society, Vol. 358, No. 5, 2005, pp. 2011-2055.
http://dx.doi.org/10.1090/S0002-9947-05-04016-X
[2] X. Luo and P. Niu, “Eigenvalues Problems for Square Sum Operators Consisting of Vector Fields,” Mathematica Applicata, Vol. 10, No. 4, 1997, pp. 101-104.
[3] E. M. Harrell and L. Hermi, “On Riesz Means of Eigenvalues,” Communications in Partial Differential Equations, Vol. 36, No. 9, 2011, pp. 1521-1543.
http://dx.doi.org/10.1080/03605302.2011.595865
[4] Yu. G. Safarov, “Riesz Means of the Distribution Function of the Eigenvalues of an Elliptic Operator,” Journal of Soviet Mathematics, Vol. 49, No. 5, 1990, pp. 1210-1212. http://dx.doi.org/10.1007/BF02208718
[5] B. Helffer and D. Robert, “Riesz Means of Bound States and Semicalassical Limit Connected with a Lieb-Thirring’s Conjecture,” Asymptotic Analysis, Vol. 3, No. 2, 1990, pp. 91-103.
[6] E. M. Harrell II and L. Hermi, “Differential Inequalities for Riesz Means and Weyl-Type Bounds for Eigenvalues,” Journal of Functional Analysis, Vol. 254, No. 12, 2008, pp. 3171-3191.
http://dx.doi.org/10.1016/j.jfa.2008.02.016
[7] G. Jia, J. Wang and Y. Xiong, “On Riesz Inequalities for Subelliptic Laplacian,” Applied Mathematics, Vol. 2, No. 6, 2011, pp. 694-698.
http://dx.doi.org/10.4236/am.2011.26091
[8] M. S. Ashbaugh and R. D. Benguria, “More Bounds on Eigenvalues Ratios for Dirichlet Laplacians in n Dimensions,” SIAM Journal on Mathematical Analysis, Vol. 24, 1993, pp. 1622-1651. http://dx.doi.org/10.1137/0524091
[9] E. M. Harrell II and J. Stubbe, “On Trace Identities and Universal Eigenvalues Estimates for Some Partial Differential Operators,” Transactions of the American Mathematical Society, Vol. 349, No. 5, 1997, pp pp. 1797-1809. http://dx.doi.org/10.1090/S0002-9947-97-01846-1
[10] P. Lvy-Bruhl, “Rsolubilit Locale et Global d’Opra-Teurs Invariants du Second Order sur des Group de Lie Nilpotents,” Bulletin des Sciences Mathématiques, Vol. 104, No. 2, 1980, pp. 369-391.
[11] A. Laptev and T. Weidl, “Sharp Lieb-Thirring Inequalities in High Dimensions,” Acta Mathematica, Vol. 184, No. 1, 2000, pp. 87-111.
http://dx.doi.org/10.1007/BF02392782

  
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