Skip to main content
SpringerLink
Log in

Weighted weak-type (1, 1) estimates for radial Fourier multipliers via extrapolation theory

  • Published:
Journal d'Analyse Mathématique Aims and scope

Abstract

In this paper, we prove a weighted estimate for the Bochner–Riesz operator at the critical index that is stronger than the weak-type (1,1) for A1 weights, in the sense that the latter can be obtained via extrapolation arguments from the former. In addition, this estimate can be transferred to averages in order to deduce weighted weak-type (1,1) results for general radial Fourier multipliers.

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. A. Carbery, G. Gasper and W. Trebels, Radial Fourier multipliers of L p(ℝ2), Proc. Natl. Acad. Sci. USA 81 (1984), 3254–3255.

    Article  Google Scholar 

  2. M. J. Carro, From restricted weak-type to strong-type estimates, J. LondonMath. Soc. 70 (2004), 750–762.

    Article  MathSciNet  MATH  Google Scholar 

  3. M. J. Carro, L. Grafakos and J. Soria, Weighted weak-type (1, 1) estimates via Rubio de Francia extrapolation, J. Funct. Anal. 269 (2015), 1203–1233.

    Article  MathSciNet  MATH  Google Scholar 

  4. M. J. Carro and J. Soria, Restricted weak-type Rubio de Francia extrapolation for p > p 0 with applications to exponential integrability estimates, Adv. Math 290 (2016), 888–918.

    Article  MathSciNet  MATH  Google Scholar 

  5. M. Christ, Weak type (1, 1) bounds for rough operators, Ann. of Math. 128 (1988), 19–42.

    Article  MathSciNet  MATH  Google Scholar 

  6. M. Christ, Lp bounds for spectral multipliers on nilpotent groups, Trans. Amer. Math. Soc. 328 (1991), 73–81.

    MathSciNet  MATH  Google Scholar 

  7. D. Cruz-Uribe, J. M. Martell and C. Pérez, Weights, Extrapolation and the Theory of Rubio de Francia, Operator Theory, Advances and Applications, Vol. 215, Birkhäuser, Basel, 2011.

    MATH  Google Scholar 

  8. D. Cruz-Uribe and C. J. Neugebauer, The structure of the reverse Hölder classes, Trans. Amer. Math. Soc. 347 (1995), 2941–2960.

    MathSciNet  MATH  Google Scholar 

  9. H. Dappa and W. Trebels, On maximal functions generated by Fourier multipliers, Ark. Mat. 23 (1985), 241–259.

    Article  MathSciNet  MATH  Google Scholar 

  10. J. Duoandikoetxea, Fourier Analysis, Graduate Studies in Mathematics, Vol. 29, American Mathematical Society, Providence, RI, 2001.

  11. J. Duoandikoetxea, Extrapolation of weights revisited: new proofs and sharp bounds, J. Funct. Anal. 260 (2011), 1886–1901.

    Article  MathSciNet  MATH  Google Scholar 

  12. J. Duoandikoetxea, A. Moyua, O. Oruetxebarria and E. Seijo, Radial Ap weights with applications to the disc multiplier and the Bochner–Riesz operators, Indiana Univ. Math. J. 57 (2008), 1261–1281.

    Article  MathSciNet  MATH  Google Scholar 

  13. G. Gasper and W. Trebels, A characterization of localized Bessel potential spaces and applications to Jacobi and Hankel multipliers, Studia Math. 65 (1979), 243–278.

    Google Scholar 

  14. L. Grafakos, Classical and Modern Fourier Analysis, Graduate Texts in Mathematics, Vol. 249, Springer, New York, 2008.

  15. L. Hörmander, Estimates for translation invariant operators in Lp spaces, ActaMath. 104 (1960), 93–139.

    MATH  Google Scholar 

  16. R. Kerman and A. Torchinsky, Integral inequalities with weights for the Hardy maximal function, Studia Math. 71 (1982), 277–284.

    Article  MathSciNet  MATH  Google Scholar 

  17. D. Kurtz, Littlewood–Paley and multiplier theorems on weighted Lp spaces, Trans. Amer. Math. Soc. 259 (1980), 235–254.

    MathSciNet  MATH  Google Scholar 

  18. D. Kurtz and R. Wheeden, Results on weighted norm inequalities for multipliers, Trans. Amer. Math. Soc. 255 (1979), 343–362.

    Article  MathSciNet  MATH  Google Scholar 

  19. M. Lacey, An elementary proof of the A2 Bound, Israel J. Math. 217 (2017), 181–195.

    Article  MathSciNet  MATH  Google Scholar 

  20. B. Muckenhoupt, Weighted norm inequalities for the Hardymaximal function, Trans. Amer. Math. Soc. 165 (1972), 207–226.

    Google Scholar 

  21. C. Pérez, Singular integrals and weights, inHarmonic and Geometric Analysis, Advanced Courses in Mathematics - CRM Barcelona, Birkhaüser, Basel, 2015, pp. 91–143.

    Book  MATH  Google Scholar 

  22. A. Seeger, Estimates near L1 for Fourier multipliers and maximal functions, Arch. Math. 53 (1989), 188–193.

    Article  MathSciNet  MATH  Google Scholar 

  23. A. Seeger, Notes on Radial and Quasiradial Fourier Multipliers, https://doi.org/www.math.wisc.edu/~seeger/papers/trest-padova.pdf

  24. X. Shi and Q. Sun, Weighted norm inequalities for Bochner–Riesz operators and singular integral operators, Proc. Amer. Math. Soc. 116 (1992), 665–673.

    Article  MathSciNet  MATH  Google Scholar 

  25. T. Tao and J. Wright, Endpoint multiplier theorems of Marcinkiewicz type, Rev. Mat. Iberoam. 17 (2001), 521–558.

    Article  MathSciNet  MATH  Google Scholar 

  26. W. Trebels, Some necessary conditions for radial Fourier multipliers, Proc. Amer. Math. Soc. 58 (1976), 97–103.

    Article  MathSciNet  MATH  Google Scholar 

  27. W. Trebels, Multipliers for (C, α)-Bounded Fourier Expansions in Banach Spaces and Approximation Theory, Lecture Notes in Mathematics, Vol. 329, Springer, New York, 1973.

  28. A. Vargas, Weighted weak type (1,1) bounds for rough operators, J. LondonMath. Soc. 54 (1996), 297–310.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departament de Matematiques i Informatica, Universitat de Barcelona, 08007, Barcelona, Spain

    María J. Carro & Carlos Domingo-Salazar

Authors
  1. María J. Carro
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Carlos Domingo-Salazar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to María J. Carro.

Additional information

The authors were supported by grants MTM2013-40985-P and 2014SGR289.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Carro, M.J., Domingo-Salazar, C. Weighted weak-type (1, 1) estimates for radial Fourier multipliers via extrapolation theory. JAMA 138, 83–105 (2019). https://doi.org/10.1007/s11854-019-0018-6

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11854-019-0018-6

Search

Navigation