Tempered distributions and Fourier transform on the Heisenberg group
Résumé
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 function
to 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.
Origine : Fichiers produits par l'(les) auteur(s)
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