Endpoint Estimates of Commutators of Singular Integrals vs. Conditions on the Symbol

  • Elida Ferreyra
  • Guillermo FloresEmail author
  • Mauricio Ramseyer
  • Beatriz Viviani


Endpoint estimates of commutators \(T_b\) of singular integrals T, are studied over general spaces that include, in particular, BMO and Lipschitz spaces. We also characterize the conditions of the symbol b in order to obtain the boundedness of the commutators of all Riesz transforms between these spaces. Finally, we apply these results over several interesting known spaces.


Commutators Singular integrals General weighted Lipschitz and BMO spaces Weighted inequalities 

Mathematics Subject Classification

30H35 42B25 42B35 



We express ours thanks to the referees for their careful reading and their useful suggestions to improve the presentation of this paper.


  1. 1.
    Bramanti, M., Brandolini, L., Viviani, B.: Global \(W^{2, p}\)-estimates for nondivergence elliptic operators with potentials satisfying a reverse Hölder condition. Ann. Mat. Pur. Appl. 191(2), 339–362 (2012)CrossRefGoogle Scholar
  2. 2.
    Cardoso, I., Viola, P., Viviani, B.: Interior \(L^p\)-estimates and Local \(A_p\)-weights. Revista de la UMA, PrepintGoogle Scholar
  3. 3.
    Chang, D.-C., Li, S.-Y.: On the boundedness of multipliers, commutators and the second derivatives of Green’s operators on \(H^1\) and \(BMO\). Ann. Scuola Norm. Sup. Pisa 28, 341–356 (1999)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Chiarenza, F., Frasca, M., Longo, P.: Interior \(W^{2,p}\)-estimates for nondivergence elliptic equations with discontinuous coefficients. Ricerche di Mat. XL, 149–168 (1991)Google Scholar
  5. 5.
    Chiarenza, F., Frasca, M., Longo, P.: Interior \(W^{2, p}\)-solvability of the Dirichlet problem for nondivergence elliptic equations with \(VMO\) coefficients. Trans. Am. Math Soc. 336, 841–853 (1993)zbMATHGoogle Scholar
  6. 6.
    Coifman, R., Rochberg, R., Weiss, G.: Factorization theorems for Hardy spaces in several variables. Ann. Math. 103, 611–635 (1976)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Franchi, B., Pérez, C., Wheeden, R.L.: Self-improving properties of John-Niremberg and Poincaré inequalities on spaces of homogeneous type. J. Funct. Anal. 153, 108–146 (1998)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Harboure, E., Segovia, C., Torrea, J.l: Boundedness of commutators of fraccional and singular integrals for the extreme values of \(p\). Ill. J. Math. 41, 676–700 (1997)CrossRefGoogle Scholar
  9. 9.
    Harboure, E., Salinas, O., Viviani, B.: Boundedness of the fractional integral on weighted Lebesgue and Lipschitz spaces. Trans. Am. Math. J. 349(1), 235–255 (1997)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Harboure, E., Salinas, O., Viviani, B.: A look at \(BMO (\omega )\) through measures Carleson. J. Fourier Anal. Appl. 13(3), 267–284 (2007)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Janson, S.: On functions with conditions on the mean oscillation. Ark. Mat. 14(1–2), 189–196 (1976)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Li, S.Y.: Toeplitz operators on Hardy space \(H^p(S)\) with \(0<p\le 1\). Integral Equ. Operator Theory 15, 802–824 (1992)CrossRefGoogle Scholar
  13. 13.
    Muckenhoupt, B., Wheeden, R.L.: Weighted bounded mean oscilation and the Hilbet transform. Stud. Math. 54 221-237, (1975/76)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Nakai, E.: Pointwise multipliers for functions of weighted bounded mean oscillations. Stud. Math. 105, 105–119 (1993)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Nakai, E.: Singular and fractional integral operators on Campanato spaces with variable growth conditions. Rev. Mat. Complut. 23, 355–381 (2010)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Nakai, E., Yabuta, K.: Pointwise multipliers for functions of bounded mean oscillations. J. Math. Soc. Jpn. 37, 207–218 (1985)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Ramseyer, M.: Operadores en espacios de Lebesgue generalizados. Ph.D. thesis, Universidad Nacional del Litoral, (2013).
  18. 18.
    Ramseyer, M., Salinas, O., Viviani, B.: Lipschitz type smoothness of the fractional integral on variable exponent spaces. Math. Anal. Appl. 403, 95–106 (2013)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Ramseyer, M., Salinas, O., Viviani, B.: Fractional intergral and Riesz transform acting on certain Lipschitz spaces. Mich. Math. J. 67, 35–56 (2016)CrossRefGoogle Scholar
  20. 20.
    Spanne, S.: Some functions spaces defined using the mean oscilation over cubes. Ann. Scuola Norm. Sup. Pisa 19, 593–608 (1965)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Stegenga, D.A.: Bounded Toeplitz operators on \(H^1\) and applications of duality between \(H^1\) and functions of mean bounded oscillation. Am. J. Math. 98(3), 573–589 (1976)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Zhong Sun, Y., Su, W.: An endpoint estimate for the commutator of singular integrals. Acta Math. Sin. 21, 1249–1258 (2005)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Zhong Sun, Y., Su, W.: Interior \(H^1\)-estimates for second order elliptic equations with vanishing \(LMO\) coefficients. J. Funct. Anal. 235, 235–260 (2006)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2020

Authors and Affiliations

  • Elida Ferreyra
    • 1
  • Guillermo Flores
    • 1
    Email author
  • Mauricio Ramseyer
    • 2
  • Beatriz Viviani
    • 2
  1. 1.CIEM-FaMAF, Universidad Nacional de CórdobaCórdobaArgentina
  2. 2.IMAL (CONICET-UNL) and FIQ (UNL)Paraje El PozoArgentina

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