Hilbert–Schimdt and Trace Class Pseudo-Differential Operators and Weyl Transforms on the Affine Group

Abstract

We give necessary and sufficient conditions on the symbols for which the corresponding pseudo-differential operators on the affine group are Hilbert–Schimdt operators. We also give a characterization of trace class pseudo-differential operators on the affine group. A trace formula for these trace class operators is also obtained. We have also obtained the \(L^2\) boundedness of the Weyl transforms on the affine group.

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Correspondence to M. W. Wong.

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The research of M. W. Wong has been supported by the Natural Sciences and Engineering Research Council of Canada under Discovery Grant 0008562. The research of Aparajita Dasgupta has been supported by Science and Engineering Research Board (SERB) under the MATRICS Grant, MTR 2019/001426.

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Dasgupta, A., Nayak, S.K. & Wong, M.W. Hilbert–Schimdt and Trace Class Pseudo-Differential Operators and Weyl Transforms on the Affine Group. J. Pseudo-Differ. Oper. Appl. 12, 11 (2021). https://doi.org/10.1007/s11868-021-00380-4

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Keywords

  • Affine group
  • Duflo–Moore operators
  • Fourier transform
  • Hilbert–Schimdt operator
  • Trace class operator
  • Trace
  • Fourier–Wigner transform
  • Wigner transform
  • Weyl transform

Mathematics Subject Classification

  • Primary 47G30