Let H₁ be the 3-dimensional Heisenberg group. We prove that a modified version of the spherical transform is an isomorphism between the space 𝓢ₘ(H₁) of Schwartz functions of type m and the space 𝓢(Σₘ) consisting of restrictions of Schwartz functions on ℝ² to a subset Σₘ of the Heisenberg fan with |m| of the half-lines removed. This result is then applied to study the case of general Schwartz functions on H₁.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm214-3-1,
author = {Francesca Astengo and Bianca Di Blasio and Fulvio Ricci},
title = {Fourier transform of Schwartz functions on the Heisenberg group},
journal = {Studia Mathematica},
volume = {215},
year = {2013},
pages = {201-222},
zbl = {1277.43012},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm214-3-1}
}
Francesca Astengo; Bianca Di Blasio; Fulvio Ricci. Fourier transform of Schwartz functions on the Heisenberg group. Studia Mathematica, Tome 215 (2013) pp. 201-222. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm214-3-1/