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

Abstract

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.

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References

  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)

  2. 2.

    Cardoso, I., Viola, P., Viviani, B.: Interior \(L^p\)-estimates and Local \(A_p\)-weights. Revista de la UMA, Prepint

  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)

  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)

  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)

  6. 6.

    Coifman, R., Rochberg, R., Weiss, G.: Factorization theorems for Hardy spaces in several variables. Ann. Math. 103, 611–635 (1976)

  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)

  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)

  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)

  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)

  11. 11.

    Janson, S.: On functions with conditions on the mean oscillation. Ark. Mat. 14(1–2), 189–196 (1976)

  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)

  13. 13.

    Muckenhoupt, B., Wheeden, R.L.: Weighted bounded mean oscilation and the Hilbet transform. Stud. Math. 54 221-237, (1975/76)

  14. 14.

    Nakai, E.: Pointwise multipliers for functions of weighted bounded mean oscillations. Stud. Math. 105, 105–119 (1993)

  15. 15.

    Nakai, E.: Singular and fractional integral operators on Campanato spaces with variable growth conditions. Rev. Mat. Complut. 23, 355–381 (2010)

  16. 16.

    Nakai, E., Yabuta, K.: Pointwise multipliers for functions of bounded mean oscillations. J. Math. Soc. Jpn. 37, 207–218 (1985)

  17. 17.

    Ramseyer, M.: Operadores en espacios de Lebesgue generalizados. Ph.D. thesis, Universidad Nacional del Litoral, (2013). http://bibliotecavirtual.unl.edu.ar:8080/tesis/handle/11185/445

  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)

  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)

  20. 20.

    Spanne, S.: Some functions spaces defined using the mean oscilation over cubes. Ann. Scuola Norm. Sup. Pisa 19, 593–608 (1965)

  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)

  22. 22.

    Zhong Sun, Y., Su, W.: An endpoint estimate for the commutator of singular integrals. Acta Math. Sin. 21, 1249–1258 (2005)

  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)

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Acknowledgements

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

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Correspondence to Guillermo Flores.

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This research was partially supported by grants from CONICET (Argentina), SeCyT (Universidad Nacional de Córdoba) and the Universidad Nacional del Litoral.

Communicated by Winfried Sickel.

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Ferreyra, E., Flores, G., Ramseyer, M. et al. Endpoint Estimates of Commutators of Singular Integrals vs. Conditions on the Symbol. J Fourier Anal Appl 26, 12 (2020) doi:10.1007/s00041-019-09714-9

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Keywords

  • Commutators
  • Singular integrals
  • General weighted Lipschitz and BMO spaces
  • Weighted inequalities

Mathematics Subject Classification

  • 30H35
  • 42B25
  • 42B35