Skip to main content
SpringerLink
Log in

Linear operators, Fourier integral operators and k-plane transforms on rearrangement-invariant quasi-Banach function spaces

  • Published:
Positivity Aims and scope Submit manuscript

Abstract

We establish the mapping properties of linear operators on rearrangement-invariant quasi-Banach function spaces. Our result applies to those linear operators that map \(L^{p}\) to \(L^{q}\) with \(p\not =q\). Therefore, it can be used to study the mapping properties of the fractional integral operators, the Fourier integral operators and the k-plane transforms on rearrangement-invariant quasi-Banach function spaces.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Beals, M.: \(L^{p}\) boundedness of Fourier integral operators. In: Memoirs of the AMS, 264. American Mathematical Society (1982)

  2. Bennett, C., Sharpley, R.: Interpolations of Operators. Academic Press, Cambridge (1988)

    Google Scholar 

  3. Bergh, J., Löfström, J.: Interpolation Spaces. Springer, Berlin (1976)

    Google Scholar 

  4. Bourbaki, N.: Intégration Livre VI. Hermann, Paris (1963)

    Google Scholar 

  5. Brudnyi, Y., Krugljak, N.: Interpolation Functors and Interpolation Spaces, vol. I. North-Holland, Amsterdam (1991)

    Google Scholar 

  6. Brenner, P.: \((L^{p}, L^{p^{\prime }})\) estimates for Fourier integral operators related to hyperbolic equations. Math. Z. 152, 273–286 (1977)

    MathSciNet  Google Scholar 

  7. Curbera, G., García-Cuerva, J., Martell, J., Pérez, C.: Extrapolation with weights, rearrangement-invariant function spaces, modular inequalities and applications to singular integrals. Adv. Math. 203, 256–318 (2006)

    MathSciNet  Google Scholar 

  8. Dmitriev, V.I., Krein, S.G.: Interpolation of operators of weak type. Anal. Math. 4, 83–99 (1978)

    MathSciNet  Google Scholar 

  9. Drury, S.: \(L^{p}\) estimates for the \(X\)-ray transform. Ill. J. Math. 27, 125–129 (1983)

    Google Scholar 

  10. Drury, S.: Generalizations of Riesz potentials and \(L^{p}\) estimates for certain \(k\)-plane transforms. Ill. J. Math. 28, 495–512 (1984)

    Google Scholar 

  11. Duistermaat, J., Hor̈mander, L.: Fourier integral operators II. Acta Math. 128, 183–269 (1972)

    MathSciNet  Google Scholar 

  12. Duistermaat, J.: Fourier Integral Operators. Courant Institute of Mathematical Sciences, New York University, New York (1973)

    Google Scholar 

  13. Edmund, D.E., Kerman, R., Pick, L.: Optimal Sobolev imbeddings involving rearrangement-invariant quasinorms. J. Funct. Anal. 170, 307–355 (2000)

    MathSciNet  Google Scholar 

  14. Edmunds, D.E., Evans, W.D.: Hardy Operators, Function Spaces and Embeddings. Springer, Berlin (2004)

    Google Scholar 

  15. Egorov, Y.: On canonical transformations of pseudo-differential operators (Russian). Uspehi Mat. Nauk. 24, 235–236 (1969)

    MathSciNet  Google Scholar 

  16. Gogatishvilli, A., Opic, B., Trebels, W.: Limiting reiteration for real interpolation with slowly varying functions. Math. Nachr. 278, 86–107 (2005)

    MathSciNet  Google Scholar 

  17. Gustavsson, J., Peetre, J.: Interpolation of Orlicz spaces. Studia Math. 60, 33–59 (1977)

    MathSciNet  Google Scholar 

  18. Gustavsson, J.: A function parameter in connection with interpolation of Banach spaces. Math. Scand. 42, 289–305 (1978)

    MathSciNet  Google Scholar 

  19. Ho, K.-P.: Littlewood-Paley spaces. Math. Scand. 108, 77–102 (2011)

    MathSciNet  Google Scholar 

  20. Ho, K.-P.: Fourier integrals and Sobolev embedding on rearrangement-invariant quasi-Banach function spaces. Ann. Acad. Sci. Fenn. Math. 41, 897–922 (2016)

    MathSciNet  Google Scholar 

  21. Ho, K.-P.: Hardy’s inequalities on Hardy spaces. Proc. Jpn. Acad. Ser. A Math. Sci. 92, 125–130 (2016)

    Google Scholar 

  22. Ho, K.-P.: Fourier type transforms on rearrangement-invariant quasi-Banach function spaces. Glasgow Math. J. 61, 231–248 (2019)

    MathSciNet  Google Scholar 

  23. Ho, K.-P.: Interpolation of sublinear operators which map into Riesz spaces and applications. Proc. Am. Math. Soc. 147, 3479–3492 (2019)

    MathSciNet  Google Scholar 

  24. Holmstedt, T.: Interpolation of quasi-normed spaces. Math. Scand. 26, 177–199 (1970)

    MathSciNet  Google Scholar 

  25. Hor̈mander, L.: Fourier integral operators. Acta Math. 127, 79–183 (1971)

    MathSciNet  Google Scholar 

  26. Hor̈mander, L.: The analysis of linear partial differential operators III–IV. In: Grundlehren der Mathematischen Wissenschaften, pp. 274–275. Springer

  27. Kalton, N., Peck, N., Roberts, J.: An \(F\)-space sampler. In: London Mathematical Society, Lecture Note Series, vol. 89. Cambridge University Press (1984)

  28. Kapitanskii, L.: Some generalizations of Strichartz–Brenner inequality. Leningr. Math. J. 1, 693–726 (1990)

    MathSciNet  Google Scholar 

  29. Krein, S.G., Petunin, Y.I., Semenov, E.M.: Interpolation of Linear Operators. American Mathematical Society, Providence, RI (1982)

    Google Scholar 

  30. Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces I and II. Springer, Berlin (1996)

    Google Scholar 

  31. Montgomery-Smith, S.: The Hardy Operator and Boyd Indices, Interaction between Functional analysis, Harmonic analysis, and Probability. Marcel Dekker Inc., New York (1996)

    Google Scholar 

  32. Nirenberg, L., Treves, F.: On local solvability of linear partial differential equations. Commun. Pure Appl. Math. 24, 279–288 (1971)

    Google Scholar 

  33. Oberlin, D., Stein, E.: Mapping properties of the Radon transform. Indiana J. Math. 31, 641–650 (1982)

    MathSciNet  Google Scholar 

  34. Okada, S., Ricker, W., Sánchez Pérez, E.: Optimal Domain and Integral Extension of Operators. Birkhäuser, Basel (2008)

    Google Scholar 

  35. Phong, D.H., Stein, E.: Models of degenerate Fourier integral operators and Radon transforms. Ann. Math. 140, 703–722 (1994)

    MathSciNet  Google Scholar 

  36. Prince, J., Links, J.: Medical Imaging Signals and Systems. Pearson Prentice Hall, Upper Saddle River (2006)

    Google Scholar 

  37. Rubin, B.: Introduction to Radon Transforms, Encyclopedia of Mathematics and Applications, vol. 160. Cambridge University Press, Cambridge (2015)

    Google Scholar 

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

    MathSciNet  Google Scholar 

  39. Seeger, A.: Degenerate Fourier integral operators in the plane. Duke Math. J. 71, 685–745 (1993)

    MathSciNet  Google Scholar 

  40. Seeger, A.: Radon transforms and finite type condition. J. Am. Math. Soc. 11, 869–897 (1998)

    MathSciNet  Google Scholar 

  41. Sogge, C.: Fourier integrals in classical analysis. In: Cambridge Tracts in Mathematics, vol. 105. Cambridge University Press (1993)

  42. Stein, E.: Harmonic Analysis. Princeton University Press, Princeton (1993)

    Google Scholar 

  43. Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam (1978)

    Google Scholar 

  44. Taylor, M.: Pseudodifferential Operators. Princeton Universiyt Press, Princeton (1981)

    Google Scholar 

  45. Treves, F.: Introduction to Pseudo-Differential and Fourier Integral Operators, vol. 1 and 2. Plenum Press, New York (1982)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Information Technology, The Education University of Hong Kong, 10 Lo Ping Road, Tai Po, Hong Kong, China

    Kwok-Pun Ho

Authors
  1. Kwok-Pun Ho
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kwok-Pun Ho.

Additional information

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

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ho, KP. Linear operators, Fourier integral operators and k-plane transforms on rearrangement-invariant quasi-Banach function spaces. Positivity 25, 73–96 (2021). https://doi.org/10.1007/s11117-020-00750-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11117-020-00750-0

Keywords

Mathematics Subject Classification

Search

Navigation