Skip to main content

Global Fourier Integral Operators in the Plane and the Square Function


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.

This is a preview of subscription content, access via your institution.


  1. Asada, K., Fujiwara, D.: On some oscillatory integral transformations in \(L^2(R^n)\). Jpn. J. Math. (N.S.) 4(2), 299–361 (1978)

    Article  Google Scholar 

  2. Beals, R.M.: \(L^p\) Boundedness of Fourier Integral Operators. Mem. Am. Math. Soc. (264). American Mathematical Society, Providence (1982)

    Google Scholar 

  3. Bergh, J., Löfström, J.: Interpolation Spaces, An Introduction. Springer, Grundlehren der Mathematischen Wissenschaften (1976)

    Book  Google Scholar 

  4. Castro, A.J., Israelsson, A., Staubach, W.: Regularity of Fourier integral operators with amplitudes in general Hörmander classes. Anal. Math. Phys. 121, 11 (2021)

    MATH  Google Scholar 

  5. Cordoba, A.: A note on Bochner–Riesz operators. Duke Math. J. 36(3), 505–511 (1979)

    MathSciNet  MATH  Google Scholar 

  6. Cordoba, A.: Geometric Fourier analysis. Ann. Inst. Fourier (Grenoble) 32(3), 215–226 (1982)

    MathSciNet  Article  Google Scholar 

  7. Coriasco, S., Ruzhansky, M.: On the boundedness of Fourier integral operators on \(L^p(\mathbb{R} ^n)\). C. R. Acad. Sci. Paris Ser. I 348, 847–851 (2010)

    Article  Google Scholar 

  8. Coriasco, S., Ruzhansky, M.: Global \(L^p\) continuity of Fourier integral operators. Trans. Am. Math. Soc. 366(5), 2575–2596 (2014)

    Article  Google Scholar 

  9. Cuerva, J.G., Rubio de Francia, J.L.: Weighted Norm Inequalities and Related Topics, North-Holland Mathematics Studies, 116. North-Holland, Amsterdam (1985)

    MATH  Google Scholar 

  10. Dos Santos Ferreira, D., Staubach, W.: Global and Local Regularity of Fourier Integral Operators on Weighted and Unweighted Spaces. Mem. Am. Math. Soc. 229 (2014)

  11. Duoandikoetxea, J.: Fourier Analysis. Graduate Studies in Mathematics. American Mathematical Society, Providence (2001)

    Google Scholar 

  12. Éskin, G.I.: Degenerate elliptic pseudo-differential equations of principal type (Russian). Mat. Sb. (N.S.) 82(124), 585–628 (1970)

    MathSciNet  Google Scholar 

  13. Folland, G.B.: Real Analysis. Modern Techniques and their Applications, Pure and Applied Mathematics. A Wiley-Interscience Publication. Wiley, New York (1984)

    Google Scholar 

  14. Hörmander, L.: Fourier integral operators, I. Acta Math. 127, 79–183 (1971)

    MathSciNet  Article  Google Scholar 

  15. Manna, R., Ratnakumar, P.K.: Local Smoothing of Fourier integral operators and hermite functions. In: Georgiev V., Ozawa T., Ruzhansky M., Wirth J. (eds) Advances in Harmonic Analysis and Partial Differential Equations, pp 1–35, Trends in Mathematics. Birkhäuser, Cham. (2020)

  16. Miyachi, A.: On some estimates for the wave equation in \(L^p\) and \(H^p\). J. Fac. Sci. Univ. Tokyo Sect. IA Math 27(2), 331–354 (1980)

    MathSciNet  MATH  Google Scholar 

  17. Mockenhaupt, G., Seeger, A., Sogge, C.D.: Wave front sets, local smoothing and Bourgain’s circular maximal theorem. Ann. Math. (2) 136(1), 207–218 (1992)

    MathSciNet  Article  Google Scholar 

  18. Peral, J.: \(L^p\) estimates for the wave equation. J. Funct. Anal. 36, 114–145 (1980)

    Article  Google Scholar 

  19. Ruzhansky, M., Sugimoto, M.: Global \(L^2\) boundedness theorems for a class of Fourier integral operators. Commun. PDE 31(4–6), 547–569 (2006)

    Article  Google Scholar 

  20. Seeger, A., Sogge, C.D., Stein, E.M.: Regularity properties of Fourier integral operators. Ann. Math. (2) 134(2), 231–251 (1991)

    MathSciNet  Article  Google Scholar 

  21. Sogge, C.D.: Fourier Integrals in Classical Analysis, Cambridge Tracts in Math., 105, Cambridge University Press, Cambridge (1993)

  22. Sogge, C.D.: Propagation of singularities and maximal functions in the plane. Invent. Math. 104, 349–376 (1991)

    MathSciNet  Article  Google Scholar 

  23. Stein, E.M.: Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, 30. Princeton Univeristy Press, Princeton (1970)

    Google Scholar 

  24. Stein, E.M.: Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, 43, Monographs in Harmonic Analysis. III. Princeton University Press, Princeton, NJ (1993)

    Google Scholar 

  25. Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series, 32. Princeton University Press, Princeton (1971)

    Google Scholar 

Download references


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.

Author information

Authors and Affiliations


Corresponding author

Correspondence to P. K. Ratnakumar.

Additional information

Communicated by Fabio Nicola.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Manna, R., Ratnakumar, P.K. Global Fourier Integral Operators in the Plane and the Square Function. J Fourier Anal Appl 28, 25 (2022).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI:


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

Mathematics Subject Classification

  • Primary 35S30
  • Secondary 42B25
  • 42B37