The Equivalence of Certain Norms on the Heisenberg Group

DOI: 10.4236/apm.2013.36073   PDF   HTML     3,435 Downloads   5,889 Views  

Abstract

Let IHn be the (2n+1)-dimensional Heisenberg group. In this paper, we shall give among other things, the properties of some homogeneous norms relative to dilations on the IHn and prove the equivalence of these norms.

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M. Egwe, "The Equivalence of Certain Norms on the Heisenberg Group," Advances in Pure Mathematics, Vol. 3 No. 6, 2013, pp. 576-578. doi: 10.4236/apm.2013.36073.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] R. Howe, “On the Role of the Heisenberg Group in Harmonic Analysis,” Bulletin of the American Mathematical Society, Vol. 3, 1980, pp. 821-843. doi:10.1090/S0273-0979-1980-14825-9
[2] D. Muller, “Analysis of Invariant PDO’s on the Heisenberg Group,” ICMS-Instructional Conference, Edinburg, 6-16 April 1999, pp. 1-23.
[3] M. E. Egwe, “Aspects of Harmonic Analysis on the Heisenberg Group,” Ph.D. Thesis, University of Ibadan, Ibadan, 2010.
[4] A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, “Stratified Lie Groups and Potential Theory for Their Sub-Laolacians (Springer Monographs in Mathematics),” Springer, Berlin, 2007.
[5] G. B. Folland and E. M. Stein, “Estimate for the Complex and Analysis on the Heisenberg Group,” Communications on Pure and Applied Mathematics, Vol. 27, No. 4, 1974, pp. 429-522. doi:10.1002/cpa.3160270403
[6] G. B. Folland, “A Fundamental Solution for a Subelliptic Operator,” Bulletin of the American Mathematical Society, Vol. 79, No. 2, 1973, p. 373. doi:10.1090/S0002-9904-1973-13171-4
[7] J. R. Lee and A. Naor, “Lp Metrics on the Heisenberg Group and the Geomans-Linial Conjecture,” Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science, Berkeley, 21-24 October 2006, pp. 99-108.
[8] J. Cygan, “Subadditivity of Homogeneous Norms on Certain Nilpotent Lie Groups,” Proceedings of the American Mathematical Society, Vol. 83, 1981, pp. 69-70. doi:10.1090/S0002-9939-1981-0619983-8
[9] N. Laghi and N. Lyall, “Strongly Singular Integral Operators Associated to Different Quasi-Norms on the Heisenberg Group,” Mathematical Research Letters, Vol. 14, No. 5, 2007, pp. 825-238.
[10] S. G. Krantz, “Explorations in Harmonic Analysis with Applications to Complex Function Theory and the Heisenberg Group,” Birkhauser, Boston, 2009.

  
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