Fourier transform of Schwartz functions on the Heisenberg group
Volume 214 / 2013
Studia Mathematica 214 (2013), 201-222
MSC: Primary 43A80; Secondary 22E25.
DOI: 10.4064/sm214-3-1
Abstract
Let $H_1$ be the 3-dimensional Heisenberg group. We prove that a modified version of the spherical transform is an isomorphism between the space $\mathcal S_m(H_1)$ of Schwartz functions of type $m$ and the space $\mathcal S(\varSigma _m)$ consisting of restrictions of Schwartz functions on $\mathbb R^2$ to a subset $\varSigma _m$ 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 $H_1$.
