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Global Fourier Integral Operators in the Plane and the Square Function

Abstract

We prove the local smoothing estimate for general Fourier integral operators with phase function of the form \(\phi (x,t,\xi )=x\cdot \xi + t \, q(\xi )\), with \(q \in C^\infty ( {\mathbb {R}}^2 \setminus \{0\} )\), homogeneous of degree one, and amplitude functions in the symbol class of order \(m \le 0\). The result is global in the space variable, and also improves our previous work in this direction (Manna et al (in: Georgiev et al., Advances in harmonic analysis and partial differential equations, Trends in Mathematics. Birkhäuser, Cham, pp. 1–35, 2020)). The approach involves a reduction to operators with amplitude function depending only on the covariable, and a new estimate for square function based on angular decomposition.

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Acknowledgements

The authors wish to thank the Harish-Chandra Research institute, Dept. of Atomic Energy, Govt. of India, for providing excellent research facility. We also wish to thank the referees for their comments and suggestions, which helped us to improve the paper.

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Correspondence to P. K. Ratnakumar.

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Communicated by Fabio Nicola.

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Manna, R., Ratnakumar, P.K. Global Fourier Integral Operators in the Plane and the Square Function. J Fourier Anal Appl 28, 25 (2022). https://doi.org/10.1007/s00041-022-09916-8

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  • DOI: https://doi.org/10.1007/s00041-022-09916-8

Keywords

  • Fourier integral operator
  • Wave front set
  • Local smoothing
  • Square function

Mathematics Subject Classification

  • Primary 35S30
  • Secondary 42B25
  • 42B37