J. Aust. Math. Soc.  81 (2006), 297–319
The Gelfand transform of homogeneous distributions on Heisenberg type groups

Francesca Astengo
  Dipartimento di Matematica
  Università di Genova
  16146 Genova
  Italia
  astengo@dima.unige.it
and
Bianca Di Blasio
  Dipartimento di Matematica
  Università di Roma "Tor Vergata"
  00133 Roma
  Italia
  diblasio@mat.uniroma2.it


Abstract
A distribution on a Heisenberg type group of homogeneous dimension $Q$ is a biradial kernel of type $\alpha$ if it coincides with a biradial function, homogeneous of degree $\alpha-Q$, and smooth away from the identity. We prove that a distribution is a biradial kernel of type $\alpha$, $0 \leq \alpha < Q$, 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 $-\alpha/2$. A similar result holds for radial kernels on the Heisenberg group.
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