Journal of Fourier Analysis and Applications

, Volume 19, Issue 4, pp 712–730 | Cite as

Characterization of the Weak-Type Boundedness of the Hilbert Transform on Weighted Lorentz Spaces

  • Elona Agora
  • María J. Carro
  • Javier Soria


We characterize the weak-type boundedness of the Hilbert transform H on weighted Lorentz spaces \(\varLambda^{p}_{u}(w)\), with p>0, in terms of some geometric conditions on the weights u and w and the weak-type boundedness of the Hardy–Littlewood maximal operator on the same spaces. Our results recover simultaneously the theory of the boundedness of H on weighted Lebesgue spaces L p (u) and Muckenhoupt weights A p , and the theory on classical Lorentz spaces Λ p (w) and Ariño-Muckenhoupt weights B p .


Weighted Lorentz spaces Hilbert transform Muckenhoupt weights Bp weights 

Mathematics Subject Classification (2010)

26D10 42A50 



We would like to thank the referee for some useful comments which have improved the final version of this paper.

The first author would also like to thank the State Scholarship Foundation I.K.Y., of Greece. H oλoϰλήρωση της εργασίας αυτής έγιϛε στo πλαίσιo της υλoπoίησης τoυ μεταπτυχιαϰoύ πρoγράμματoς πoυ συγχρηματoδoτήϑηϰε μέσω της Πράξης “Πρóγραμμα χoρήγησης υπoτρoφιώϛ I.K.ϒ. με διαδιϰασία εξατoμιϰευμέϛης αξιoλóγησης αϰαδ. έτoυς 2011–2012” απó πóρoυς τoυ E.Π. “Eϰπαίδευση ϰαι δια βίoυ μάϑηση” τoυ Eυρωπαϊϰoύ ϰoιϛωϛιϰoύ ταμείoυ (EKT) ϰαι τoυ EΣΠA, τoυ 2007–2013.


  1. 1.
    Agora, E., Carro, M.J., Soria, J.: Boundedness of the Hilbert transform on weighted Lorentz spaces. J. Math. Anal. Appl. 395, 218–229 (2012) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Andersen, K.F.: Weighted generalized Hardy inequalities for nonincreasing functions. Can. J. Math. 43, 1121–1135 (1991) CrossRefGoogle Scholar
  3. 3.
    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) zbMATHGoogle Scholar
  4. 4.
    Bagby, R., Kurtz, D.S.: A rearranged good λ inequality. Trans. Am. Math. Soc. 293(1), 71–81 (1986) MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bennett, C., Sharpley, R.: Interpolation of Operators. Pure and Applied Mathematics, vol. 129. Academic Press, Boston (1988) zbMATHGoogle Scholar
  6. 6.
    Carro, M.J., Raposo, J.A., Soria, J.: Recent developments in the theory of Lorentz spaces and weighted inequalities. In: Mem. Amer. Math. Soc, vol. 187 (2007). No. 877 Google Scholar
  7. 7.
    Cerdà, J., Martín, J.: Interpolation restricted to decreasing functions and Lorentz spaces. Proc. Edinb. Math. Soc. 42(2), 243–256 (1999) zbMATHCrossRefGoogle Scholar
  8. 8.
    Chung, H., Hunt, R., Kurtz, D.S.: The Hardy–Littlewood maximal function on L(p,q) spaces with weights. Indiana Univ. Math. J. 31, 109–120 (1982) MathSciNetCrossRefGoogle Scholar
  9. 9.
    Coifman, R., Fefferman, C.: Weighted norm inequalities for maximal functions and singular integrals. Stud. Math. 51, 241–250 (1974) MathSciNetzbMATHGoogle Scholar
  10. 10.
    García Cuerva, J., Rubio de Francia, J.L.: Weighted Norm Inequalities and Related Topics. North-Holland Mathematics Studies, vol. 116. North-Holland, Amsterdam (1985). Notas de Matemática [Mathematical Notes], vol. 104 zbMATHCrossRefGoogle Scholar
  11. 11.
    Grafakos, L.: Classical Fourier Analysis, 2nd edn. Graduate Texts in Mathematics, vol. 249. Springer, New York (2008) zbMATHGoogle Scholar
  12. 12.
    Hunt, R., Muckenhoupt, B., Wheeden, R.: Weighted norm inequalities for the conjugate function and Hilbert transform. Trans. Am. Math. Soc. 176, 227–251 (1973) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Hytönen, T., Pérez, C.: Sharp weighted bounds involving A . To appear in Anal. PDE.
  14. 14.
    Journé, J.-L.: Calderón-Zygmund Operators, Pseudodifferential Operators and the Cauchy Integral of Calderón. Lecture Notes in Mathematics, vol. 994. Springer, Berlin (1983) Google Scholar
  15. 15.
    Lorentz, G.: Some new functional spaces. Ann. Math. 51, 37–55 (1950) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Lorentz, G.: On the theory of spaces Λ. Pac. J. Math. 1, 411–429 (1951) MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Muckenhoupt, B.: Weighted norm inequalities for the Hardy maximal function. Trans. Am. Math. Soc. 165, 207–226 (1972) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Neugebauer, C.J.: Weighted norm inequalities for averaging operators of monotone functions. Publ. Mat. 35(2), 429–447 (1991) MathSciNetzbMATHGoogle Scholar
  19. 19.
    Neugebauer, C.J.: Some classical operators on Lorentz spaces. Forum Math. 4(3), 135–146 (1992) MathSciNetzbMATHGoogle Scholar
  20. 20.
    Sawyer, E.: Boundedness of classical operators on classical Lorentz spaces. Stud. Math. 96, 145–158 (1990) MathSciNetGoogle Scholar
  21. 21.
    Soria, J.: Lorentz spaces of weak-type. Q. J. Math. 49(193), 93–103 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Wilson, J.M.: Weighted inequalities for the dyadic square function without dyadic A . Duke Math. J. 55, 19–50 (1987) Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Departament de Matemàtica Aplicada i AnàlisiUniversitat de BarcelonaBarcelonaSpain

Personalised recommendations