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On the Composition of Rough Singular Integral Operators

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Abstract

In this paper, we investigate the behavior of the bounds of the composition for rough singular integral operators on the weighted space. More precisely, we obtain the quantitative weighted bounds of the composite operator for two singular integral operators with rough homogeneous kernels on \(L^p({\mathbb {R}}^d,\,w)\), \(p\in (1,\,\infty )\), which is smaller than the product of the quantitative weighted bounds for these two rough singular integral operators. Moreover, at the endpoint \(p=1\), the \(L\log L\) weighted weak-type bound is also obtained, which has interests of its own in the theory of rough singular integral even in the unweighted case.

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References

  1. Benea, C., Bernicot, F.: Conservation de certaines propriétés á travers un contrôle épars d’un opérateur et applications au projecteur de Leray–Hopf. arXiv:1703.00228

  2. Calderón, A.P.: Algebras of singular integral operators. In: 1968 Proceedings of the International Congress Mathematicians, pp. 393-395. Izdat. “Mir”, Moscow (1966)

  3. Calderón, A.P.: Algebras of singular integral operators. Singular integrals. In: Proceedings of Symposia in Pure Mathematics, Chicago, 1966, pp. 18–55. American Mathematical Society, Providence, RI (1967)

  4. Calderón, A.P., Zygmund, A.: On the existence of certain singular integrals. Acta Math. 88, 85–139 (1952)

    Article  MathSciNet  Google Scholar 

  5. Calderón, A.P., Zygmund, A.: On singular integrals. Am. J. Math. 78, 289–309 (1956)

    Article  Google Scholar 

  6. Calderón, A.P., Zygmund, A.: Algebras of certain singular operators. Am. J. Math. 78, 310–320 (1956)

    Article  MathSciNet  Google Scholar 

  7. Carozza, N., Passarelli di Napoli, A.: Composition of maximal operators. Publ. Mat. 40, 397–409 (1996)

    Article  MathSciNet  Google Scholar 

  8. Conde-Alonso, J.M., Culiuc, A., Di Plinio, F., Ou, Y.: A sparse domination principle for rough singular integrals. Anal. PDE 10, 1255–1284 (2017)

    Article  MathSciNet  Google Scholar 

  9. Christ, M.: Inversion in some algebras of singular integral operators. Rev. Mat. Iberoamericana 4(2), 219–225 (1988)

    Article  MathSciNet  Google Scholar 

  10. Christ, M., Rubio de Francia, J.-L.: Weak type (1, 1) bounds for rough operators, II. Invent. Math. 93, 225–237 (1988)

    Article  MathSciNet  Google Scholar 

  11. Connett, W.C.: Singular integrals near \(L^1\). In: Wainger, S., Weiss, G. ( eds.) Proceedings of Symposia in Pure Mathematics of American Mathematical Society, vol. 35 I, pp. 63–165 (1979)

  12. Di Plinio, F., Hytönen, T., Li, K.: Sparse bounds for maximal rough singular integrals via the Fourier transform. arXiv:1706.09064

  13. Fan, D., Pan, Y.: Singular integral operators with rough kernels supported by subvarieties. Am. J. Math. 119, 799–839 (1997)

    Article  MathSciNet  Google Scholar 

  14. Grafakos, L.: Classic fourier analysis. In: Axler, S., Ribe, K. (eds.) Graduate Texts in Mathematics, 3rd edn. Springer, New York (2014)

    Google Scholar 

  15. Grafakos, L.: Modern fourier analysis. In: Axler, S., Ribe, K. (eds.) Graduate Texts in Mathematics, 3rd edn. Springer, New York (2014)

    Google Scholar 

  16. Grafakos, L., Stefanov, A.: \(L^p\) bounds for singular integrals and maximal singular integrals with rough kernels. Indiana Univ. Math. J. 47, 455–469 (1998)

    Article  MathSciNet  Google Scholar 

  17. Hu, G.: Weighted vector-valued estimates for a non-standard Calderón–Zygmund operator. Nonlinear Anal. 165, 143–162 (2017)

    Article  MathSciNet  Google Scholar 

  18. Hu, G.: Weighted weak type endpoint estimates for the composition of Calderón–Zygmund operators. J. Aust. Math. Soc. (2018). https://doi.org/10.1017/S1446788719000107

  19. Hu, G.: Quantitative weighted bounds for the composition of Calderón–Zygmund operators. Banach J. Math. Anal. 13, 133–150 (2019)

    Article  MathSciNet  Google Scholar 

  20. Hu, G., Li, D.: A Cotlar type inequality for the multilinear singular integral operators and its applications. J. Math. Anal. Appl. 290, 639–653 (2004)

    Article  MathSciNet  Google Scholar 

  21. Hytönen, T., Pérez, C.: Sharp weighted bounds involving \(A_1\). Anal. PDE 6, 777–818 (2013)

    Article  MathSciNet  Google Scholar 

  22. Hytönen, T., Pérez, C.: The \(L(\log L)^{\epsilon }\) endpoint estimate for maximal singular integral operators. J. Math. Anal. Appl. 428, 605–626 (2015)

    Article  MathSciNet  Google Scholar 

  23. Hytönen, T., Lacey, M.T., Pérez, C.: Sharp weighted bounds for the \(q\)-variation of singular integrals. Bull. Lond. Math. Soc. 45, 529–540 (2013)

    Article  MathSciNet  Google Scholar 

  24. Hytönen, T., Roncal, L., Tapiola, O.: Quantitative weighted estimates for rough homogeneous singular integrals. Israel J. Math. 218, 133–164 (2017)

    Article  MathSciNet  Google Scholar 

  25. Lerner, A.K.: A note on weighted bounds for rough singular integrals. Comptes Rend. Math. 356, 77–80 (2018)

    Article  MathSciNet  Google Scholar 

  26. Lerner, A.K.: A weak type estimates for rough singular integrals. Rev. Mat. Iberoam. arXiv:1705.07397

  27. Lerner, A.K., Ombrosi, S., Pérez, C.: \(A_1\) bounds for Calderón–Zygmund operators related to a problem of Muckenhoupt and Wheeden. Math. Res. Lett. 16, 149–156 (2009)

    Article  MathSciNet  Google Scholar 

  28. Li, K., Pérez, C., Rivera-Rios, I.P., Roncal, L.: Weighted norm inequalities for rough singular integral operators. J. Geom. Anal. 29, 2526–2564 (2019)

    Article  MathSciNet  Google Scholar 

  29. Nagel, A., Ricci, F., Stein, E.M., Wainger, S.: Algebras of singular integral operators with kernels controlled by multiple norms. Mem. Am. Math. Soc. 256(1230), vii+141 (2018)

    MathSciNet  MATH  Google Scholar 

  30. Oberlin, R.: Estimates for compositions of maximal operators with singular integrals. Can. Math. Bull. 56, 801–813 (2013)

    Article  MathSciNet  Google Scholar 

  31. Phone, D.H., Stein, E.M.: Some further classes of pseudo-differential and singular integral operators arising in boundary-value problems I, composition of operators. Am. J. Math. 104, 141–172 (1982)

    Article  Google Scholar 

  32. Ricci, F., Weiss, G.: A characterization of \(H^1(S^{n-1})\). In: Wainger, S., Weiss, G. (eds.) Proceedings of Symposia in Pure Mathematics, vol. 35 I, pp. 289–294. American Mathematical Society (1979)

  33. Seeger, A.: Singular integral operators with rough convolution kernels. J. Am. Math. Soc. 9, 95–105 (1996)

    Article  MathSciNet  Google Scholar 

  34. Strichartz, R.S.: Compositions of singular integral operators. J. Funct. Anal. 49, 91–127 (1982)

    Article  MathSciNet  Google Scholar 

  35. Wilson, M.J.: Weighted inequalities for the dyadic square function without dyadic \(A_{\infty }\). Duke Math. J. 55, 19–50 (1987)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors would like to thank Dr. Kangwei Li for his helpful comments and suggestions.

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Authors and Affiliations

  1. School of Applied Mathematics, Beijing Normal University, Zhuhai, 519087, People’s Republic of China

    Guoen Hu

  2. Institute for Advanced Study in Mathematics, Harbin Institute of Technology, Harbin, 150001, People’s Republic of China

    Xudong Lai

  3. School of Mathematics, Beijing Normal university, Beijing, 100875, People’s Republic of China

    Qingying Xue

Authors
  1. Guoen Hu
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  2. Xudong Lai
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  3. Qingying Xue
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Correspondence to Xudong Lai.

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The research of the first author was supported by NSFC (No. 11871108). The research of the second author was supported by National Natural Science Foundation of China (No. 11801118), China Postdoctoral Science Foundation (Nos. 2017M621253, 2018T110279), and the Fundamental Research Funds for the Central Universities. The research of the third author was supported partly by NSFC (Nos. 11471041, 11671039, 11871101) and NSFC-DFG (No. 11761131002).

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Hu, G., Lai, X. & Xue, Q. On the Composition of Rough Singular Integral Operators. J Geom Anal 31, 2742–2765 (2021). https://doi.org/10.1007/s12220-020-00374-6

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