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A characterisation of the Fourier transform on the Heisenberg group. (English) Zbl 1260.43010

An operator that takes the convolution product into the pointwise product is essentially the Fourier transform. Taking this characterization of the Fourier transform as a model, analogous results are obtained in this paper for the Weyl transform and the group Fourier transform on the Heisenberg group \(\mathbb H^n\): 1. It is proved that a continuous homomorphism of the algebra \(L^1(\mathbb C^n)\) with twisted convolution multiplication into the operator algebra \(\mathcal B(L^2(\mathbb R^n))\) is ‘essentially’ the Weyl transform. 2. A characterization, in the same spirit, of the group Fourier transform on the Heisenberg group, as a map \(L^1(\mathbb H^n) \rightarrow L^{\infty}(\mathbb R^*, \mathcal S_2, \mu)\) is obtained. Here \(\mathcal S_2\) denotes the algebra of Hilbert-Schmidt operators on \(L^2(\mathbb R^n)\) and the measure \(\mu\) on \(\mathbb R^*\) is defined by \(d\mu(\lambda) = (2\pi)^{-n-1}|\lambda|^n d\lambda\).

MSC:

43A32 Other transforms and operators of Fourier type
42A85 Convolution, factorization for one variable harmonic analysis
43A80 Analysis on other specific Lie groups
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