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On the anti-Wick symbol as a Gelfand-Shilov generalized function. (English) Zbl 1505.47051

It is known that the Weyl symbol of a pseudo-differential operator can be deduced from the anti-Wick symbol by the convolution with the Gaussian. The authors present here a result in the opposite direction. Namely, given the Weyl symbol in the space of the Schwartz distributions \(\mathcal{S}'\), they prove that the anti-Wick symbol of the same operator can be expressed as an ultradistribution of Gelfand-Shilov type. By duality, the proof is reduced to evaluate the regularity and the asymptotic behaviour of the functions obtained from the Schwartz test space \(\mathcal{S}\) by means of the convolution with the Gaussian. Relevant references are [N. Lerner, Metrics on the phase space and non-selfadjoint pseudo-differential operators. Basel: Birkhäuser (2010; Zbl 1186.47001)] for the definition of the anti-Wick pseudo-differential operators and [F. Nicola and L. Rodino, Global pseudo-differential calculus on Euclidean spaces. Basel: Birkhäuser (2010; Zbl 1257.47002)] for properties of Gelfand-Shilov classes.

MSC:

47G30 Pseudodifferential operators
46F05 Topological linear spaces of test functions, distributions and ultradistributions
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References:

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