Revisiting the Fourier transform on the Heisenberg group. (English) Zbl 1285.43003

A theorem of S. Alesker et al. [American Mathematical Society 226; Advances in the Mathematical Sciences 63, 11–26 (2009; Zbl 1184.42009)] characterizes the Fourier transform on \(\mathbb{R}^n\) as essentially the only transform on the space of tempered distributions which interchanges convolutions and pointwise products.
In this paper the authors obtain a similar characterization for the Fourier transform on the Heisenberg group \(\mathbb{H}^n = \mathbb{C}^n \times \mathbb{R}\) equipped with the group law \[ (z,t)(w,s)=(z+w, t+s+\frac{1}{2}\operatorname{Im} (z\cdot\overline{w})). \] The group Fourier transform on \(\mathbb{H}^n\) is defined to be the operator valued function \[ \pi_{\lambda}(f)=\int_{\mathbb{H}^n}f(z,t)\pi_{\lambda}(z,t)\;dz dt \] where \(\pi_{\lambda}(z,t)\) is the Schrödinger representation of \(\mathbb{H}^n\) for each \(\lambda \in \mathbb{R}^{*}=\mathbb{R} \setminus \{ 0\}\) and the Fourier-Weyl transform on \(\mathbb{H}^n\) is defined to be the operator valued function \[ \hat{f}(\lambda, \zeta)=\int_{\mathbb{H}^n}f(z,t)\pi_{\lambda, \zeta}(z,t)\;dz dt \] where \(\pi_{\lambda,\zeta}(z,t)=\chi_{\zeta}(z,t)\pi_{\lambda}(z,t)\) is the representation of \(\mathbb{H}^n\) parametrised by \((\lambda, \zeta) \in \mathbb{R}^{*}\times \mathbb{C}^n\) and \(\chi_{\zeta}(z,t)=e^{i\operatorname{Re} z\cdot\overline{\zeta}}\) is a character. The convolutions \(f*g\) and \(f*_{3}g\) on \(\mathbb{H}^n\) are defined by \[ (f*g)(z,t)=\int_{\mathbb{H}^n}f((z,t)(-w,-s))g(w,s)dwds, \]
\[ f*_{3}g(z,t)=\int_{-\infty}^{\infty}f(z,t-s)g(z,s)\;ds \] and a translation on \(\mathbb{H}^n\) is given by \[ (R_{(z,t)}f)(w,s)=f((w,s)(z,t)). \] Let \(\mathcal{S}(\mathbb{H}^n)\) stand for the Schwartz space on \(\mathbb{H}^n\). Denote by \(\hat{\mathcal{S}}(\mathbb{H}^n)\) the image of \(\mathcal{S}(\mathbb{H}^n)\) under the group Fourier transform \(f \rightarrow \pi_{\lambda}(f)\).
{Theorem. } Let \(T: \mathcal{S}(\mathbb{H}^n) \rightarrow \hat{\mathcal{S}}(\mathbb{H}^n)\) be a bijection which satisfies
(i) \(T(f\ast g)=TfTg\) and
(ii) \(T(f*_{3}g)(\lambda)=2\pi\int_{\mathbb{C}^n}\mathcal{F}^{-1}f(w,\lambda) Tg(\lambda, w)\;dw\)
where \(\mathcal{F}^{-1}\) stands on the inverse Euclidean Fourier transform on \(\mathbb{C}^n\times \mathbb{R}\) and \(Tg(\lambda, w)\) stands for \(Tg(\lambda, w)=\pi_{\lambda}(-i\lambda^{-1}w,0)Tg(\lambda)\pi_{\lambda}(i\lambda^{-1}w,0)\).
Then there exists a map \(\zeta: \mathbb{R}^{*} \rightarrow \mathbb{C}^n\) such that \(Tf(\lambda)=\hat{f}(\lambda, \zeta(\lambda))\).
Corollary. If in addition to the hypothesis of the Theorem the map T
satisfies \(T(R_{(z,0)}f)(\lambda)=Tf(\lambda)\pi_{\lambda}(z)^{*}\). Then \(Tf(\lambda)=\pi_{\lambda}(f)\).


43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
43A80 Analysis on other specific Lie groups
22E30 Analysis on real and complex Lie groups
35K08 Heat kernel


Zbl 1184.42009
Full Text: DOI Euclid