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The Gabor wave front set in spaces of ultradifferentiable functions

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Abstract

We consider the spaces of ultradifferentiable functions \(\mathcal {S}_\omega \) as introduced by Björck (and its dual \(\mathcal {S}'_\omega \)) and we use time-frequency analysis to define a suitable wave front set in this setting and obtain several applications: global regularity properties of pseudodifferential operators of infinite order and the micro-pseudolocal behaviour of partial differential operators with polynomial coefficients and of localization operators with symbols of exponential growth. Moreover, we prove that the new wave front set, defined in terms of the Gabor transform, can be described using only Gabor frames. Finally, some examples show the convenience of the use of weight functions to describe more precisely the global regularity of (ultra)distributions.

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Acknowledgements

The authors are very grateful to the reviewers for the careful reading and for the comments and remarks which improve the presentation and the quality of the paper.

The authors were partially supported by the INdAM-Gnampa Project 2016 “Nuove prospettive nell’analisi microlocale e tempo-frequenza”, by FAR 2013, FAR 2014 (University of Ferrara) and by the project “Ricerca Locale - Analisi di Gabor, operatori pseudodifferenziali ed equazioni differenziali” (University of Torino). The research of the second author was partially supported by the project MTM2016-76647-P.

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Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università di Ferrara, Via Machiavelli n. 30, 44121, Ferrara, Italy

    Chiara Boiti

  2. Instituto Universitario de Matemática Pura y Aplicada IUMPA, Universitat Politècnica de València, Camino de Vera, s/n, 46071, Valencia, Spain

    David Jornet

  3. Dipartimento di Matematica, Università di Torino, Via Carlo Alberto n. 10, 10123, Turin, Italy

    Alessandro Oliaro

Authors
  1. Chiara Boiti
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  2. David Jornet
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  3. Alessandro Oliaro
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Correspondence to Alessandro Oliaro.

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Communicated by K. Gröchenig.

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Boiti, C., Jornet, D. & Oliaro, A. The Gabor wave front set in spaces of ultradifferentiable functions. Monatsh Math 188, 199–246 (2019). https://doi.org/10.1007/s00605-018-1242-3

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