Some Characterisations of the Strichartz Fourier Transform on the Heisenberg Group

Abstract

Recently, Thangavelu introduced the Strichartz Fourier transform on the Heisenberg group. Around the same time, Krukowski characterised the classical Fourier transform on various settings, assuming a priori that the map is an integral transform, based on structural properties and a normalisation. In this paper, we obtain analogous results for the Strichartz Fourier transform, without assuming the characterised map is an integral transform. The spectral property is formulated jointly for the sublaplacian and central vector field, and the translation property uses a twisted form of left translation. The joint spectral property, the twisted translation property and a normalisation at one explicit test function, together characterise the transform on each ray of the Heisenberg fan. The twisted translation property, paired with a Dirac delta property, gives another characterisation. We characterise the radial kernel on each ray by a Laguerre-type differential equation. This gives a characterisation of the Hankel transform.

arXiv comment20 pages