Vol. 231, No. 1, 2007

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Gutzmer’s formula and Poisson integrals on the Heisenberg group

Sundaram Thangavelu

Vol. 231 (2007), No. 1, 217–237

In 1978 M. Lassalle obtained an analogue of the Laurent series for holomorphic functions on the complexification of a compact symmetric space and proved a Plancherel type formula for such functions. In 2002 J. Faraut established such a formula, which he calls Gutzmer’s formula, for all noncompact Riemannian symmetric spaces. This was immediately put into use by B. Krotz, G. Olafsson and R. Stanton to characterise the image of the heat kernel transform. In this article we prove an analogue of Gutzmer’s formula for the Heisenberg motion group and use it to characterise Poisson integrals associated to the sublaplacian. We also use the Gutzmer’s formula to study twisted Bergman spaces.

Heisenberg group, sublaplacian, Fourier transform, Poisson integrals, Gutzmer’s formula, Laguerre functions
Mathematical Subject Classification 2000
Primary: 43A85, 43A90
Secondary: 22E25, 22E30
Received: 3 March 2006
Revised: 4 June 2006
Accepted: 21 June 2006
Published: 1 May 2007
Sundaram Thangavelu
Department of Mathematics
Indian Institute of Science
Bangalore 560 012