A characterisation of the Weyl transform (Q2882744)

A characterization of the Euclidean Fourier transform due to \textit{S. Alesker et al.} [Linear and complex analysis, Amer. Math. Soc. Transl. Ser. 2, 226, Amer. Math. Soc., Providence, RI, 11--26 (2009; Zbl 1184.42009)] says the following:NEWLINENEWLINEAssume that \(T:~S(\mathbb R^n) \rightarrow S(\mathbb R^n)\) is a bijection which admits a bijective extension (denoted by \(T\) again) from \(S'(\mathbb R^n)\) onto itself such that \(\forall f \in S(\mathbb R^n)\) and \(\varphi \in S'(\mathbb R^n),\) we have NEWLINE\[NEWLINE T(f \ast \varphi) = T(f) T(\varphi)~\text{and}~T(f\varphi) = T(f) \ast T(\varphi).NEWLINE\]NEWLINE Then, \(T\) is essentialy the Fourier transform \(\mathcal{F},\) that is, for some \(B \in \text{GL}_n(\mathbb R)\) with \(|\text{det} B| = 1,\) we have either NEWLINE\[NEWLINETf = \mathcal{F} (f~ \text{o}~ B) ~\text{or}~Tf = \overline{\mathcal{F}(f ~\text{o}~B)}.NEWLINE\]NEWLINE The paper under review is concerned with a similar characterization problem for the Weyl transform. Recall the Schrödinger representations of the Heisenberg group given by \( \pi_\lambda(z, t) = e^{i\lambda t} \pi_\lambda(z)\) where NEWLINE\[NEWLINE \pi_\lambda (z) \varphi (\xi) = e^{i\lambda (x \cdot \xi + \frac{1}{2} x \cdot y)} \varphi(\xi + y),NEWLINE\]NEWLINE \(\varphi \in L^2(\mathbb R^n)\) and \( z = x+iy \in \mathbb C^n.\) The Weyl transform of a suitable function \(f\) is defined by NEWLINE\[NEWLINE W(f) = \int_{\mathbb C^n}~f(z)~\pi(z)~dz,NEWLINE\]NEWLINE where \(\pi(z) = \pi_1(z).\) It is known that \(W(f \times g) = W(f) W(g)\) where \(f \times g\) is the twisted convolution of \(f\) and \(g\) defined by NEWLINE\[NEWLINEf \times g (z) = \int_{\mathbb C^n}~f(z-w)~g(w)~e^{i \text{Im} z \cdot \bar{w}}~dw.NEWLINE\]NEWLINE Let \(S_{op}\) be the image of \(S(\mathbb C^n)\) under the Weyl transform and \(S_{op}'\) its topological dual. For any operator \(T \in S_{op}',\) define the Fourier-Weyl transform of \(T\) to be NEWLINE\[NEWLINE \widetilde{T}(\zeta) = \pi(-2i\zeta) T \pi(2i\zeta),NEWLINE\]NEWLINE for \(\zeta = \xi+ i \eta \in \mathbb C^n.\)NEWLINENEWLINEThe main theorem in the paper is as follows: Let \(T: S(\mathbb C^n) \rightarrow S_{op}\) be a bijection which has a bijective extension (again denoted by \(T\)) from \(S'(\mathbb C^n)\) onto \(S'_{op}\) such that for all \(f \in S'(\mathbb C^n)\) and \(g \in S(\mathbb C^n),\) we have NEWLINE\[NEWLINE T(f \times g ) = T(f) T(g)~\text{and}~\widetilde{T(gf)}(0) = \mathcal{F}(f) \ast \widetilde{T(g)}(0).NEWLINE\]NEWLINE Then \(T(f) = \widetilde{W(f)}(\zeta)\) for some \(\zeta \in \mathbb C^n.\)