Hostname: page-component-76fb5796d-skm99 Total loading time: 0 Render date: 2024-04-25T13:26:02.814Z Has data issue: false hasContentIssue false
  • English
  • Français

The Gelfand transform of homogeneous distributions on Heisenberg type groups

Part of: Lie groups Abstract harmonic analysis

Published online by Cambridge University Press:  09 April 2009

Francesca Astengo  and
Bianca Di Blasio
Francesca Astengo
Affiliation:
Dipartimento di MatematicaUniversità di Genova16146 GenovaItalia e-mail: astengo@dima.unige.it
Bianca Di Blasio
Affiliation:
Dipartimento di Matematica Università di Roma“Tor Vergata” 00133 RomaItalia e-mail: diblasio@mat.uniroma2.it
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A distribution on a Heisenberg type group of homogeneous dimension Q is a biradial kernel of type α if it coincides with a biradial function, homogeneous of degree α — Q, and smooth away from the identity. We prove that a distribution is a biradial kernel of type α, 0 < α < Q, if and only if its Gelfand transform, defined on the Heisenberg fan, extends to a smooth even function on the upper half plane, homogeneous of degree −α/2. A similar result holds for radial kernels on the Heisenberg group.

Keywords

heisenberg type groups, homogeneous distributions.

MSC classification

Secondary: 43A80: Analysis on other specific Lie groups 22E30: Analysis on real and complex Lie groups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 81 , Issue 3 , December 2006 , pp. 297 - 319
Copyright
Copyright © Australian Mathematical Society 2006

References

[1]Astengo, F. and Di Blasio, B., ‘The Schwartz space and homogeneous distributions on H-type groups’, Monatsh. Math. 132 (2001), 197–214.CrossRefGoogle Scholar
[2]Benson, C., Jenkins, J. and Ratcliff, G., ‘The spherical transform of a Schwartz function on the Heisenberg group’, J. Funct. Anal. 154 (1998), 379–423.CrossRefGoogle Scholar
[3]Cowling, M., ‘Unitary and uniformly bounded representations of some simple Lie groups’, in: Harmonic analysis and group representations, C.I.M.E. (Liguori, Napoli, 1982) pp. 49–128.Google Scholar
[4]Folland, G. B., ‘Subelliptic estimates and function spaces on nilpotent Lie groups’, Ark. Mat. 13 (1975), 161–207.CrossRefGoogle Scholar
[5]Folland, G. B. and Stein, E. M., ‘Estimates for the complex and analysis on the Heisenberg group’, Comm. Pure Appl. Math. 27 (1974), 429–522.CrossRefGoogle Scholar
[6]Fraser, A. J., ‘Convolution kernels of (n + 1)-fold Marcinkiewicz multipliers on the Heisenberg group’, Bull. Austral. Math. Soc. 64 (2001), 353–376.CrossRefGoogle Scholar
[7]Fraser, A. J., ‘An (n+1)-fold Marcinkiewicz multipliers theorem on the Heisenberg group’, Bull. Austral. Math. Soc. 63 (2001), 35–58.CrossRefGoogle Scholar
[8]Geller, D., ‘Local solvability and homogeneous distributions on the Heisenberg group’, Comm. Partial Differential Equations 5 (1980), 475–560.CrossRefGoogle Scholar
[9]Kaplan, A., ‘Fundamental solution for a class of hypoelliptic PDE generated by composition of quadratic forms’, Trans. Amer. Math. Soc. 258 (1980), 147–153.CrossRefGoogle Scholar
[10] Müller, D., Peloso, M. and Ricci, F., ‘L p-spectral multipliers for the Hodge Laplacian acting on 1-forms on the Heisenberg group’, preprint.Google Scholar
[11]Müller, D., Ricci, F. and Stein, E. M., ‘Marcinkiewicz multipliers and multi-parameter structure on Heisenberg (-type) groups, I’, Invent. Math. 119 (1995), 199–233.CrossRefGoogle Scholar
[12]Slome, S. E., ‘The heisenberg group and the group Fourier transform of regular homogeneous distributions’, Studia Math. 143 (2000), 251–266.CrossRefGoogle Scholar
[13]Stein, E. M., Harmonic analysis real variable methods, orthogonality, and oscillatory integrals (Princeton University Press, Princeton, 1993).Google Scholar