The Gelfand transform of homogeneous distributions of Heisenberg type groups

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