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
Bianca Di Blasio
  Dipartimento di Matematica
  Università di Roma "Tor Vergata"
  00133 Roma

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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