Journal of Fourier Analysis and Applications

, Volume 22, Issue 6, pp 1431–1439 | Cite as

Weak-Type Boundedness of the Hardy–Littlewood Maximal Operator on Weighted Lorentz Spaces

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Abstract

The main goal of this paper is to provide a characterization of the weak-type boundedness of the Hardy–Littlewood maximal operator, M, on weighted Lorentz spaces \(\Lambda ^p_u(w)\), whenever \(p>1\). This solves a problem left open in (Carro et al., Mem Am Math Soc. 2007). Moreover, with this result, we complete the program of unifying the study of the boundedness of M on weighted Lebesgue spaces and classical Lorentz spaces, which was initiated in the aforementioned monograph.

Keywords

Weighted Lorentz spaces Hardy–Littlewood maximal operator 

Mathematics Subject Classification

42B25 47A30 

Notes

Acknowledgments

We would like to thank Prof. Javier Soria for the helpful discussions related with the subject of this paper, and also for his useful comments that let us improve the presentation of this paper. The first and second authors would also like to thank the University of Barcelona (UB) and the Institute of Mathematics of the University of Barcelona (IMUB) for providing us all the facilities and hosting us during the research stay that led to this collaboration. Finally, we would like to express our thanks to the Referees for their useful comments and suggestions. This work has been partially supported by Grants MTM2012-36378, MTM2013-40985-P, 2014SGR289, CONICET-PIP 2012-2014: 11220110101018, CONICET-PIP 2009-435, UBACyT 2002013010042BA, PICT 2014-1480 and UNLP 11X681.

References

  1. 1.
    Agora, E., Antezana, J., Carro, M.J., Soria, J.: Lorentz-Shimogaki and Boyd theorems for weighted Lorentz spaces. J. Lond. Math. Soc. 89(2), 321–336 (2014)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Ariño, M.A., Muckenhoupt, B.: Maximal functions on classical Lorentz spaces and Hardy’s inequality with weights for nonincreasing functions. Trans. Am. Math. Soc. 320(2), 727–735 (1990)MATHGoogle Scholar
  3. 3.
    Bennett, C., Sharpley, R.: Interpolation of Operators, Pure and Applied Mathematics, 129. Academic Press Inc, Boston, MA (1988)MATHGoogle Scholar
  4. 4.
    Carro, M.J., Soria, J.: Weighted Lorentz spaces and the Hardy operator. J. Funct. Anal. 112(2), 480–494 (1993)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Carro, M.J., Raposo, J.A., Soria, J.: Recent developments in the theory of Lorentz spaces and weighted inequalities. Mem. Am. Math. Soc. (2007). doi:10.1090/memo/0877
  6. 6.
    Lerner, A.K., Pérez, C.: A new characterization of the Muckenhoupt \(A_p\) weights through an extension of the Lorentz-Shimogaki theorem. Indiana Univ. Math. J. 56(6), 2697–2722 (2007)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Lorentz, G.: Some new functional spaces. Ann. Math. 51(2), 37–55 (1950)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Lorentz, G.: On the theory of spaces \(\Lambda \). Pac. J. Math. 1, 411–429 (1951)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Muckenhoupt, B.: Weighted norm inequalities for the Hardy maximal function. Trans. Am. Math. Soc. 165, 207–226 (1972)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Neugebauer, C.J.: Weighted norm inequalities for averaging operators of monotone functions. Publ. Mat. 35(2), 429–447 (1991)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Elona Agora
    • 1
  • Jorge Antezana
    • 1
    • 2
  • María J. Carro
    • 3
  1. 1.Instituto Argentino de Matemática “Alberto P. Calderón”Buenos AiresArgentina
  2. 2.Department of Mathematics, Faculty of Exact SciencesNational University of La PlataLa PlataArgentina
  3. 3.Department of Applied Mathematics and AnalysisUniversity of BarcelonaBarcelonaSpain

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