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Characterization of Smooth Symbol Classes by Gabor Matrix Decay

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Abstract

For \(m\in \mathbb {R}\) we consider the symbol classes \(S^m\), \(m\in \mathbb {R}\), consisting of smooth functions \(\sigma \) on \({\mathbb {R}^{2d}}\) such that \(|\partial ^\alpha \sigma (z)|\le C_\alpha (1+|z|^2)^{m/2}\), \(z\in {\mathbb {R}^{2d}}\), and we show that can be characterized by an intersection of different types of modulation spaces. In the case \(m=0\) we recapture the Hörmander class \(S^0_{0,0}\) that can be obtained by intersection of suitable Besov spaces as well. Such spaces contain the Shubin classes \(\Gamma ^m_\rho \), \(0<\rho \le 1\), and can be viewed as their limit case \(\rho =0\). We exhibit almost diagonalization properties for the Gabor matrix of \(\tau \)-pseudodifferential operators with symbols in such classes, extending the characterization proved by Gröchenig and Rzeszotnik (Ann Inst Fourier 58(7):2279–2314, 2008). Finally, we compute the Gabor matrix of a Born–Jordan operator, which allows to prove new boundedness results for such operators.

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Acknowledgements

The authors would like to thank Fabio Nicola and S. Ivan Trapasso for fruitful conversations and comments. The authors are very grateful to the reviewers for their comments and in particular for the improvements of Proposition 3.5. The first author was partially supported by MIUR Grant Dipartimenti di Eccellenza 20182022, CUP: E11G18000350001, DISMA, Politecnico di Torino.

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  1. Department of Mathematical Sciences, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Turin, Italy

    Federico Bastianoni

  2. Department of Mathematics, University of Torino, Via Carlo Alberto 10, 10123, Turin, Italy

    Elena Cordero

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  1. Federico Bastianoni
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  2. Elena Cordero
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Correspondence to Elena Cordero.

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Communicated by Karlheinz Gröchenig.

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Bastianoni, F., Cordero, E. Characterization of Smooth Symbol Classes by Gabor Matrix Decay. J Fourier Anal Appl 28, 3 (2022). https://doi.org/10.1007/s00041-021-09895-2

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