The Equivalence of Certain Norms on the Heisenberg Group

Murphy E. Egwe^{*}

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

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.