# Tempered distributions and Fourier transform on the Heisenberg group

## TL;DR

This paper extends the Fourier transform to tempered distributions on the Heisenberg group by redefining the transform as a mapping on a new set, enabling a broader analysis similar to the Euclidean case.

## Contribution

It introduces a new approach to define the Fourier transform on the Heisenberg group for tempered distributions, overcoming previous limitations by mapping to a specially structured set.

## Key findings

- Defined Fourier transform as a mapping on  ext{H}^d with a suitable metric
- Provided explicit formulas for Fourier transforms of specific functions
- Extended the Fourier transform to tempered distributions on  ext{H}^d

## Abstract

The final goal of the present work is to extend the Fourier transform on the Heisenberg group $\H^d,$ to tempered distributions. As in the Euclidean setting, the strategy is to first show that the Fourier transform is an isomorphism on the Schwartz space, then to define the extension by duality. The difficulty that is here encountered is that the Fourier transform of an integrable function on $\H^d$is no longer a function on $\H^d$ : according to the standard definition, it is a family of bounded operators on $L^2(\R^d).$ Following our new approach in\ccite{bcdFHspace}, we here define the Fourier transform of an integrable functionto be a mapping on the set~$\wt\H^d=\N^d\times\N^d\times\R\setminus\{0\}$endowed with a suitable distance $\wh d$.This viewpoint turns out to provide a user friendly description of the range of the Schwartz space on $\H^d$ by the Fourier transform, which makes the extension to the whole set of tempered distributions straightforward. As a first application, we give an explicit formula for the Fourier transform of smooth functions on $\H^d$ that are independent of the vertical variable. We also provide other examples.