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Weighted Weak-type Estimates for Multilinear Commutators of Fractional Integrals on Spaces of Homogeneous Type

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Abstract

We obtain weighted distributional inequalities for multilinear commutators of the fractional integral on spaces of homogeneous type. The techniques developed in this work involve the behavior of some fractional maximal functions. In relation to these operators, as a main tool, we prove a weighted weak type boundedness result, which is interesting in itself.

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References

  1. Chanillo, S.: A note on commutators. Indiana Univ. Math. J., 31, 7–16 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  2. Segovia, C., Torrea, J. L.: Weighted inequalities for commutators of fractional and singular integrals. Publ. Matematiques, 35, 209–235 (1991)

    MATH  MathSciNet  Google Scholar 

  3. Bramanti, M., Cerutti, M. C.: Commutators of singular integrals and fractional integrals on homogeneous spaces, Harmonic Analysis and Operator Theory, (Caracas 1994), (Providence, RI). Contemp. Math., A.M.S., 189, 81–94 (1995)

    MathSciNet  Google Scholar 

  4. Betancor, J.: A commutator theorem for fractional integrals in spaces of homogeneous type. Internat. J. Math. & Math. Sci., 24(6), 403–418 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  5. Harboure, E., Segovia, C., Torrea, J. L.: Boundedness of commutators of fractional and singular integrals for the extreme values of p. Illinois J. of Math., 41(4), 676–700 (1997)

    MathSciNet  Google Scholar 

  6. Pérez, C., Trujillo-González, R.: Sharp weighted estimates for multilinear commutators. J. London Math. Soc., 65(2), 672–692 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  7. Macías, R., Segovia, C.: Lipschitz functions on spaces of homogeneous type. Adv. Math., 33, 257–270 (1979)

    Article  MATH  Google Scholar 

  8. Bernardis, A., Salinas, O.: Two-weighted inequalities for certain maximal fractional operators on spaces of homogeneous type. Revista de la Unión Matemática Argentina, 41(3), 61–75 (1999)

    MATH  MathSciNet  Google Scholar 

  9. Cruz Uribe, D., Fiorenza, A.: Endpoint estimates and weighted norm inequalities for commutators of fractional integrals. Publ. Mat., 47, 103–131 (2003)

    MATH  MathSciNet  Google Scholar 

  10. Ding, Y., Lu, S. Z, Zhang, P.: Weak estimates for commutators of fractional integral operators. Science in China (Series A), 44(7), 877-888 (2001)

    Article  MATH  Google Scholar 

  11. Hu, G. E., Meng, Y., Yang, D. C.: Multilinear commutators for fractional integrals in non-homogeneous spaces. Publ. Mat., 48, 335–367 (2004)

    MATH  MathSciNet  Google Scholar 

  12. Pérez, C.: Endpoint estimates for commutators of singular integrals operators. J. Funct. Anal., 127, 163–185 (1995)

    Article  Google Scholar 

  13. Pérez, C.: Sharp estimates for commutators of singular integrals via iterations of the Hardy–Littlewood maximal function. Journal of Fourier Analysis and Applications, 3(6), 743–756 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  14. Pérez, C., Pradolini, G.: Endpoint estimates for commutators of singular integrals operators. J. Funct. Anal., 127, 163–185 (1995)

    Article  Google Scholar 

  15. Pradolini, G., Salinas, O.: Maximal operators on spaces of homogeneous type. Proceedings of A.M.S., 132(2), 435–441 (2003)

    Article  MathSciNet  Google Scholar 

  16. Pérez, C., Wheeden, R.: Uncertainty principle estimates for vector fields. J. Funct. Anal., 181, 146–188 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  17. Coifman, R., Weiss, G.: Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in Math., 242, Springer-Verlag, Berlin, 1971

  18. Macías, R., Torrea, J.: L 2 and L p boundedness of singular integrals on non necessarily normalized spaces of homogeneous type, Cuadernos de Matemática y Mecánica, 1-88, PEMA-INTEC-GTM, Santa Fe (Argentina)

  19. Hardy, G. H., Littlewood, J. E., Pólya, G.: Inequalities, Cambridge University Press, Cambridge, 1952

  20. Muckenhoupt, B., Wheeden, R.: Weighted norm inequalities for the fractional integrals. Trans. AMS, 192, 261–274 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  21. Pradolini, G., Salinas, O.: Commutators of singular integrals on spaces of homogeneous type, Preprint available in http://math.unl.edu.ar/preprints/

  22. Journé, L.: Calderón–Zygmund operators, pseudo-differential operator, and the Cauchy integral of Calderón, Lecture notes in Mathematics, 994, Springer, New York, 1983

  23. Stein, E. M.: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, 43, Princeton University Press, Princeton, NJ, 1993

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Authors and Affiliations

  1. Department of Mathematics, Consejo Nacional de Investigaciones Científicas y Téecnicas de la República, Argentina

    Osvaldo Gorosito, Gladis Pradolini & Oscar Salinas

Authors
  1. Osvaldo Gorosito
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  2. Gladis Pradolini
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  3. Oscar Salinas
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Corresponding author

Correspondence to Osvaldo Gorosito.

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The authors are supported by Consejo Nacional de Investigaciones Científicas y Técnicas de la República Argentina and Universidad Nacional del Litoral

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Gorosito, O., Pradolini, G. & Salinas, O. Weighted Weak-type Estimates for Multilinear Commutators of Fractional Integrals on Spaces of Homogeneous Type. Acta Math Sinica 23, 1813–1826 (2007). https://doi.org/10.1007/s10114-005-0741-z

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  • DOI: https://doi.org/10.1007/s10114-005-0741-z

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