Skip to main content

Advertisement

SpringerLink
Log in

A lower bound for \(A_{p}\) exponents for some weighted weak-type inequalities

  • Fourier Analysis
  • Published:
The Journal of Analysis Aims and scope Submit manuscript

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

We give a weak-type counterpart of the main result in Luque et al. (Math Res Lett 22(1):183–201, 2015) which allows to provide a lower bound for the exponent of the \(A_{p}\) constant in terms of the behaviour of the unweighted inequalities when \(p\rightarrow \infty \) and when \(p\rightarrow 1^{+}\). We also provide some applications to classical operators.

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. Alvarez, J., and C. Pérez. 1994. Estimates with \(A_\infty \) weights for various singular integral operators, Bollettino U.M.I. (7) 8-A :123–133.

  2. Astala, K., T. Iwaniec, and E. Saksman. 2001. Beltrami operators in the plane. Duke Mathematical Journal 107 (1): 27–56.

    Article  MathSciNet  Google Scholar 

  3. Bagby, R.J., and D.S. Kurtz. 1985. Covering lemmas and the sharp function. Proceedings of the American Mathematical Society 93: 291–296.

    Article  MathSciNet  Google Scholar 

  4. Bagby, R.J., and D.S. Kurtz. 1986. A rearranged good-\(\lambda \) inequality. Transactions of the American Mathematical Society 293: 71–81.

    MathSciNet  MATH  Google Scholar 

  5. Buckley, S.M. 1993. Estimates for operator norms on weighted spaces and reverse Jensen inequalities. Transactions of the American Mathematical Society 340 (1): 253–272.

    Article  MathSciNet  Google Scholar 

  6. Cruz-Uribe, D., SFO, J.M. Martell and C. Pérez. 2011. Weights, Extrapolation and the Theory of Rubio de Francia, Operator Theory: Advances and Applications, vol. 215, Birkhäuser/Springer Basel AG, Basel.

  7. Duoandikoetxea, J. 2011. Extrapolation of weights revisited: new proofs and sharp bounds. Journal of Functional Analysis 260 (6): 1886–1901.

    Article  MathSciNet  Google Scholar 

  8. García-Cuerva, J., and J.L. Rubio de Francia. 1985. Weighted Norm Inequalities and Related Topics, North Holland Math. Studies, vol. 116, North Holland, Amsterdam.

  9. Hytönen, T.P. 2012. The sharp weighted bound for general Calderón–Zygmund operators. Annals of Mathematics (2) 175 (3): 1473–1506.

    Article  MathSciNet  Google Scholar 

  10. T.P. Hytönen, M.T. Lacey, H. Martikainen, T. Orponen, M.C. Reguera, E.T. Sawyer, I. Uriarte-Tuero. 2012. Weak and strong type estimates for maximal truncations of Calderón-Zygmund operators on \(A_p\) weighted spaces. Journal d'Analyse Mathématique 118 (1): 177–220.

  11. Hytönen, T.P., and C. Pérez. 2013. Sharp weighted bounds involving \(A_{\infty }\). Analysis of PDE 6 (4): 777–818.

    Article  MathSciNet  Google Scholar 

  12. Lerner, A.K., S. Ombrosi, and I.P. Rivera-Ríos. 2017. On pointwise and weighted estimates for commutators of Calderón–Zygmund operators. Advances in Mathematics 319: 153–181.

    Article  MathSciNet  Google Scholar 

  13. Luque, T., C. Pérez, and E. Rela. 2015. Optimal exponents in weighted estimates without examples. Mathematical Research Letters 22 (1): 183–201.

    Article  MathSciNet  Google Scholar 

  14. Ortiz-Caraballo, C. 2011. Quadratic \(A_1\) bounds for commutators of singular integrals with BMO functions. Indiana University Mathematics Journal 60 (6): 2107–2129.

    Article  MathSciNet  Google Scholar 

  15. Ortiz-Caraballo, C., C. Pérez and E. Rela. 2013. Improving bounds for singular operators via Sharp Reverse Hölder Inequality for \(A_{\infty }\). In Operator Theory: Advances and Applications, Advances in Harmonic Analysis and Operator Theory, eds. A. Almeida, L. Castro, F. Speck, vol. 229, 303–321. Springer Basel.

  16. Pérez, C. 2015. Singular integrals and weights. Harmonic and geometric analysis, Adv. Courses Math. CRM Barcelona, 91–143. Birkhäuser/Springer Basel AG, Basel.

  17. Petermichl, S., and A. Volberg. 2002. Heating of the Ahlfors–Beurling operator: weakly quasiregular maps on the plane are quasiregular. Duke Mathematical Journal 112 (2): 281–305.

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

Both authors are supported by the Basque Government through the BERC 2014-2017 program and by Spanish Ministry of Economy and Competitiveness MINECO through BCAM Severo Ochoa excellence accreditation SEV-2013-0323. and through the project MTM2014-53850-P. I.P.R-R. is also supported by Spanish Ministry of Economy and Competitiveness MINECO through the project MTM2012-30748.

Author information

Authors and Affiliations

  1. Departamento de Matemáticas, Universidad del País Vasco UPV/EHU, IKERBASQUE, Basque Foundation for Science, BCAM, Basque Center for Applied Mathematics, Bilbao, Spain

    Carlos Pérez

  2. Departamento de Matemáticas, Universidad del País Vasco UPV/EHU and BCAM, Basque Center for Applied Mathematics, Bilbao, Spain

    Israel P. Rivera-Ríos

Authors
  1. Carlos Pérez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Israel P. Rivera-Ríos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Carlos Pérez.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pérez, C., Rivera-Ríos, I.P. A lower bound for \(A_{p}\) exponents for some weighted weak-type inequalities. J Anal 26, 191–209 (2018). https://doi.org/10.1007/s41478-018-0137-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s41478-018-0137-y

Keywords

Mathematics Subject Classification

Search

Navigation