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Quantization of Pseudo-differential Operators on the Torus

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Abstract

Pseudo-differential and Fourier series operators on the torus \({{\mathbb{T}}^{n}}=(\Bbb{R}/2\pi\Bbb{Z})^{n}\) are analyzed by using global representations by Fourier series instead of local representations in coordinate charts. Toroidal symbols are investigated and the correspondence between toroidal and Euclidean symbols of pseudo-differential operators is established. Periodization of operators and hyperbolic partial differential equations is discussed. Fourier series operators, which are analogues of Fourier integral operators on the torus, are introduced, and formulae for their compositions with pseudo-differential operators are derived. It is shown that pseudo-differential and Fourier series operators are bounded on L 2 under certain conditions on their phases and amplitudes.

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Authors and Affiliations

  1. Department of Mathematics, Imperial College London, 180 Queen’s Gate, London, SW7 2AZ, UK

    Michael Ruzhansky

  2. Institute of Mathematics, Helsinki University of Technology, P.O. Box 1100, 02015 HUT, Helsinki, Finland

    Ville Turunen

Authors
  1. Michael Ruzhansky
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  2. Ville Turunen
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Correspondence to Michael Ruzhansky.

Additional information

Communicated by Karlheinz Gröchenig.

The first author was supported in part by the JSPS Invitational Research Fellowship and by the EPSRC grants EP/E062873/1 and EP/G007233/1.

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Ruzhansky, M., Turunen, V. Quantization of Pseudo-differential Operators on the Torus. J Fourier Anal Appl 16, 943–982 (2010). https://doi.org/10.1007/s00041-009-9117-6

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  • DOI: https://doi.org/10.1007/s00041-009-9117-6

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