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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
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