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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 $$Tf(\lambda)=\pi_{\lambda}(f)$$.

##### MSC:
 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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