Differentiation and Shift Property Characterisations of the Group Fourier Transform on the Heisenberg Group

Recently, Krukowski characterised the classical Fourier transform on the real line in terms of its behaviour with respect to differentiation. For locally compact abelian groups, he obtained a characterisation based on the behaviour of the Fourier transform under the shift. Each of these results assumed that the operator (to be characterised) is an integral transform (with a kernel satisfying suitable regularity conditions), paired with a normalisation which he refers to as the Dirac delta property. In this paper, we obtain analogous results for the group Fourier transform on the Heisenberg group. Here, the differentiation property is formulated in terms of the behaviour of the group Fourier transform with respect to the left-invariant vector fields, and the shift property is based on its behaviour under right translation. Both of these characterise the group Fourier transform on the Heisenberg group, once paired with a Dirac delta property. Our results assume that the transform is continuous and linear, and do not require it to be an integral transform a priori.