Skip to main content
Log in

Restricted summability of Fourier transforms and local Hardy spaces

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier transforms. Under some conditions on θ, it is proved that the maximal operator of the θ-means defined in a cone is bounded from the amalgam Hardy space W(h p , ) to W(L p , ). This implies the almost everywhere convergence of the θ-means in a cone for all fW(L 1, ) ⊃ L 1.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Zygmund, A.: Trigonometric Series, Cambridge Press, London, 3rd edition, 2002

    MATH  Google Scholar 

  2. Móricz, F.: The maximal Fejér operator for Fourier transforms of functions in Hardy spaces. Acta Sci. Math. (Szeged), 62, 537–555 (1996)

    MathSciNet  MATH  Google Scholar 

  3. Móricz, F.: The maximal Fejér operator is bounded from H 1(T) to L 1(T). Analysis, 16, 125–135 (1996)

    MathSciNet  MATH  Google Scholar 

  4. Weisz, F.: Summability of Multi-dimensional Fourier Series and Hardy Spaces. Mathematics and Its Applications. Kluwer Academic Publishers, Dordrecht, Boston, London, 2002

    Google Scholar 

  5. Butzer, P. L., Nessel, R. J.: Fourier Analysis and Approximation, Birkhäuser Verlag, Basel, 1971

    MATH  Google Scholar 

  6. Trigub, R. M., Belinsky, E. S.: Fourier Analysis and Approximation of Functions, Kluwer Academic Publishers, Dordrecht, Boston, London, 2004

    MATH  Google Scholar 

  7. Feichtinger, H. G., Weisz, F.: The Segal algebra S 0(ℝd) and norm summability of Fourier series and Fourier transforms. Monatshefte Math., 148, 333–349 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Feichtinger, H. G., Weisz, F.: Wiener amalgams and pointwise summability of Fourier transforms and Fourier series. Math. Proc. Camb. Phil. Soc., 140, 509–536 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Stein, E. M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, N.J., 1971

    MATH  Google Scholar 

  10. Marcinkiewicz, J., Zygmund, A.: On the summability of double Fourier series. Fund. Math., 32, 122–132 (1939)

    Google Scholar 

  11. Gát, Gy.: Pointwise convergence of cone-like restricted two-dimensional (C, 1) means of trigonometric Fourier series. J. Appr. Theory, 149, 74–102 (2007)

    Article  MATH  Google Scholar 

  12. Gát, Gy., Nagy, K.: Pointwise convergence of cone-like restricted two-dimensional Fejér means of Walsh-Fourier series. Acta Mathematica Sinica, English Series. DOI: 10.1007/s10114-010-9340-8

  13. Weisz, F.: The maximal Fejér operator of multi-dimensional Fourier transforms. East J. Appr., 4, 491–503 (1998)

    MathSciNet  MATH  Google Scholar 

  14. Goldberg, D.: A local version of real Hardy spaces. Duke Math. J., 46, 27–42 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  15. Bergh, J., Löfström, J.: Interpolation Spaces, an Introduction, Springer, Berlin, 1976

    MATH  Google Scholar 

  16. Stein, E. M.: Harmonic Analysis: Real-variable Methods, Orthogonality and Oscillatory Integrals, Princeton Univ. Press, Princeton, N.J., 1993

    MATH  Google Scholar 

  17. Lu, S.: Four Lectures on Real H p Spaces, World Scientific, Singapore, 1995

    MATH  Google Scholar 

  18. Weisz, F.: Multi-dimensional Fejér summability and local Hardy spaces. Studia Math., 194, 181–195 (2009)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ferenc Weisz.

Additional information

Supported by the Hungarian Scientific Research Funds (OTKA) No. K67642

Rights and permissions

Reprints and permissions

About this article

Cite this article

Weisz, F. Restricted summability of Fourier transforms and local Hardy spaces. Acta. Math. Sin.-English Ser. 26, 1627–1640 (2010). https://doi.org/10.1007/s10114-010-9529-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-010-9529-x

Keywords

MR(2000) Subject Classification

Navigation