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Multiple Weighted Norm Inequalities for Multilinear Strongly Singular Integral Operators with Generalized Kernels

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Abstract

Lin (Nonlinear Anal 192:111699, 2020) introduced a class of multilinear strongly singular integral operators, which have weaker smoothness condition compared to the classical multilinear singular integral operators except that they do not require the size condition. In this paper, the smoothness condition of the kernel function is further reduced of the multilinear strongly singular integral operator. We get the weighted \(L^p\) boundedness of the multilinear strongly singular integral operator with generalized kernel as well as two types of endpoint estimates, respectively. Additionally, the weighted \(L^p\) boundedness of the multilinear commutator and multilinear iterated commutator generated by BMO functions and multilinear strongly singular integral operators with generalized kernels is also established, respectively.

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References

  1. Benyi, A., Chaffee, L., Naibo, V.: Strongly singular bilinear Calderón–Zygmund operators and a class of bilinear pseudodifferential operators. J. Math. Soc. Jpn. 71, 569–587 (2019)

    Article  MATH  Google Scholar 

  2. Coifman, R.R., Meyer, Y.: On commutators of singular integrals and bilinear singular integrals. Trans. Am. Math. Soc. 212, 315–331 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  3. Coifman, R.R., Meyer, Y.: Au delà des opérateurs pseudo-différentiels. Astérisque 57, 1–185 (1978)

    MATH  Google Scholar 

  4. Coifman, R.R., Meyer, Y.: Commutateurs d’intégrales singulières et opérateurs multilinéaires Ann. Inst. Fourier (Grenoble) 28, 177–202 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  5. Duong, X.T., Grafakos, L., Yan, L.X.: Multilinear operators with non-smooth kernels and commutators of singular integrals. Trans. Am. Math. Soc. 362, 2089–2113 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Fefferman, C., Stein, E.M.: \(H^{p}\) spaces of several variables. Acta Math. 129, 137–193 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  7. Garcia-Cuerva, J., Rubio de Francia, J. L.: Weighted norm inequalities and related topics. North-Holland Mathematics Studies, vol. 116. North-Holland Publishing Co., Amsterdam (1985)

  8. Grafakos, L., Kalton, N.: Multilinear Calderón–Zygmund operators on Hardy spaces. Collect. Math. 52, 169–179 (2001)

    MathSciNet  MATH  Google Scholar 

  9. Grafakos, L., Martell, J.M.: Extrapolation of weighted norm inequalities for multivariable operators and applications. J. Geom. Anal. 14, 19–46 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Grafakos, L., Torres, R.H.: Multilinear Calderón–Zygmund theory. Adv. Math. 165, 124–164 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  11. Grafakos, L., Torres, R.H.: On multilinear singular integrals of Calderón–Zygmund type. In: Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations (El Escorial, 2000). Publ. Mat., Vol. Extra, 57–91 (2002)

  12. Lerner, A.K., Ombrosi, S., Pérez, C., Torres, R.H., Trujillo-González, R.: New maximal functions and multiple weights for the multilinear Calderón–Zygmund theory. Adv. Math. 220, 1222–1264 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lin, Y.: Multilinear theory of strongly singular Calderón–Zygmund operators and applications. Nonlinear Anal. 192, 111699 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lin, Y., Han, Y.Y.: Sharp maximal and weighted estimates for multilinear iterated commutators of multilinear strongly singular Calderón-Zygmund operators. Chin. Ann. Math. Ser. A 40, 399–416 (2019)

    MATH  Google Scholar 

  15. Lin, Y., Lu, G.Z., Lu, S.Z.: Sharp maximal estimates for multilinear commutators of multilinear strongly singular Calderón–Zygmund operators and applications. Forum Math. 31, 1–18 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lin, Y., Lu, S.Z.: Strongly singular Calderón–Zygmund operators and their commutators. Jordan J. Math. Stat. 1, 31–49 (2008)

    MATH  Google Scholar 

  17. Lin, Y., Xiao, Y.Y.: Multilinear singular integral operators with generalized kernels and their multilinear commutators. Acta Math. Sin. (Engl. Ser.) 33, 1443–1462 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lin, Y., Yan, H.H.: Multilinear strongly singular Calderón–Zygmund operators and commutators on Morrey Type spaces. Jordan J. Math. Stat. 14, 351–375 (2021)

    MathSciNet  MATH  Google Scholar 

  19. Lin, Y., Zhang, N.: Sharp maximal and weighted estimates for multilinear iterated commutators of multilinear integrals with generalized kernels. J. Inequal. Appl. 276 (2017)

  20. Lu, G.Z., Zhang, P.: Multilinear Calderón–Zygmund operators with kernels of Dini’s type and applications. Nonlinear Anal. 107, 92–117 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Pérez, C., Pradolini, G., Torres, R.H., Trujillo-González, R.: End-point estimates for iterated commutators of multilinear singular integrals. Bull. Lond. Math. Soc. 46, 26–42 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. Tang, L.: Weighted estimates for vector-valued commutators of multilinear operators. Proc. R. Soc. Edinb. Sect. A. 138, 897–921 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  23. Xue, Q.Y.: Weighted estimates for the iterated commutators of multilinear maximal and fractional type operators. Studia Math. 217, 97–122 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Yang, S.H., Lin, Y.: Multilinear strongly singular integral operators with generalized kernels and applications. AIMS Math. 6, 13533–13551 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  25. Yang, S.H., Lin, Y.: Weighted estimates for multilinear strongly singular Calderón–Zygmund operators with multiple weights. Jordan J. Math. Stat. 15, 467–495 (2022)

    MathSciNet  MATH  Google Scholar 

  26. Yang, S.H., Lin, Y.: Multilinear commutators of multilinear strongly singular integral operators with generalized kernels. Georgian Math. J. (2023)

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Authors and Affiliations

  1. School of Science, China University of Mining and Technology, Beijing, 100083, People’s Republic of China

    Bixiao Wei, Shuhui Yang & Yan Lin

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  1. Bixiao Wei
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  2. Shuhui Yang
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  3. Yan Lin
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Correspondence to Shuhui Yang.

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Communicated by Matthew Daws.

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This work was partially supported by the National Natural Science Foundation of China No.12071052 and Basic Research Foundation of China University of Mining and Technology (Beijing)– Cultivation of Top Innovative Talents for Doctoral No. BBJ2023058.

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Wei, B., Yang, S. & Lin, Y. Multiple Weighted Norm Inequalities for Multilinear Strongly Singular Integral Operators with Generalized Kernels. Bull. Iran. Math. Soc. 49, 75 (2023). https://doi.org/10.1007/s41980-023-00816-1

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  • DOI: https://doi.org/10.1007/s41980-023-00816-1

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