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L2(ℝn) boundedness for Calderón commutator with rough variable kernel

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Abstract

For b ∈ Lip(ℝn); the Calderón commutator with variable kernel is defined by

$$[b,{T_1}]f(x) = p.v.\int_{{R n}} {\frac{{\Omega (x,x - y)}}{{|x - y{| {n + 1}}}}(b(x) - b(y))f(y)dy} $$

In this paper, we establish the L2(ℝn) boundedness for [b, T1] with Ω(x, z′) ∈ L∞(ℝn)-Lq(Sn-1) (q > 2(n-1)=n) satisfying certain cancellation conditions. Moreover, the exponent q > 2(n-1)/n is optimal. Our main result improves a previous result of Calderón.

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References

  1. AL-Balushi K, AL-Salman A. Certain Lp bounds for rough singular integrals. J Math Inequal, 2014, 8: 803–822

    Article  MathSciNet  MATH  Google Scholar 

  2. AL-Salman A, Pan Y. Singular integrals with rough kernels. Canad Math Bull, 2004, 47: 3–11

    Article  MathSciNet  MATH  Google Scholar 

  3. Calderón A P. Commutators of singular integral operators. Proc Natl Acad Sci USA, 1965, 53: 1092–1099

    Article  MathSciNet  MATH  Google Scholar 

  4. Calderón A P. Cauchy integrals on Lipschitz curves and related operators. Proc Natl Acad Sci USA, 1977, 74: 1324–1327

    Article  MathSciNet  MATH  Google Scholar 

  5. Calderón A P. Commutators, singular integrals on Lipschitz curves and applications. In: Lehto O, ed. Proc Inter Congr Math, Helsinki, 1978, Vol 1. Helsinki: Acad Sci Fennica, 1980, 85–96

  6. Calderón A P, Zygmund A. On a problem of Mihlin. Trans Amer Math Soc, 1955, 78: 209–224

    Article  MathSciNet  MATH  Google Scholar 

  7. Calderón A P, Zygmund A. Singular integral operators and differential equations. Amer J Math, 1957, 79: 901–921

    Article  MathSciNet  MATH  Google Scholar 

  8. Calderón A P, Zygmund A. On singular integrals with variable kernels. Appl Anal, 1978, 7: 221–238

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen J, Fan D, Ying Y. Certain operators with rough singular kernels. Canad J Math, 2003, 55: 504–532

    Article  MathSciNet  MATH  Google Scholar 

  10. Chen Y, Ding Y. L2 boundedness for commutator of rough singular integral with variable kernel. Rev Mat Iberoam, 2008, 24: 531–547

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen Y, Ding Y. Boundedness for commutators of rough hypersingular integrals with variable kernels. Michigan Math J, 2010, 59: 189–210

    Article  MathSciNet  MATH  Google Scholar 

  12. Chen Y, Ding Y. Lp bounds for the commutators of singular integrals and maximal singular integrals with rough kernels. Trans Amer Math Soc, 2015, 3671585–1608

    Google Scholar 

  13. Chen Y, Ding Y. Necessary and suffcient conditions for the bounds of the Calderón type commutator for the Littlewood-Paley operator. Nonlinear Anal, 2016, 130: 279–297

    Article  MathSciNet  MATH  Google Scholar 

  14. Chiarenza F, Frasca M, Longo P. Interior W2;p estimates for nondivergence elliptic equations with discontinuous coeffcients. Ric Mat, 1991, 40: 149–168

    MATH  Google Scholar 

  15. Chiarenza F, Frasca M, Longo P. W2;p-solvability of the Dirichlet problem for non-divergence elliptic equations with VMO coeffcients. Trans Amer Math Soc, 1993, 336: 841–853

    MathSciNet  MATH  Google Scholar 

  16. Christ M, Duoandikoetxea J, Rubio de Francia J. Maximal operators related to the Radon transform and the Calderón-Zygmund method of rotations. Duke Math J, 1986, 53: 189–209

    Article  MathSciNet  MATH  Google Scholar 

  17. Coifman R, Meyer Y. On commutators of singular integrals and bilinear integrals. Trans Amer Math Soc, 1975, 212: 315–331

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. Cowling M, Mauceri G. Inequalities for some maximal functions I. Trans Amer Math Soc, 1985, 287: 431–455

    Article  MathSciNet  MATH  Google Scholar 

  20. Duoandikoetxea J. Fourier Analysis. Grad Stud Math, Vol 29. Providence: Amer Math Soc, 2001, 106

    Google Scholar 

  21. Duoandikoetxea J, Rubio de Francia J. Maximal and singular integrals via Fourier transform estimates. Invent Math, 1986, 84: 541–561

    Article  MathSciNet  MATH  Google Scholar 

  22. Di Fazio G, Ragusa M. Interior estimates in Morrey spaces for strong solutions to non-divergence form equations with discontinuous coeffcients. J Funct Anal, 1993, 112: 241–256

    Article  MathSciNet  MATH  Google Scholar 

  23. Ding Y, Lin C C, Shao S. On Marcinkiewicz integral with variable kernels. Indiana Univ Math J, 2004, 53: 805–821

    Article  MathSciNet  MATH  Google Scholar 

  24. Fefferman C. Recent progress in classical Fourier analysis. In: James R D, ed. Proc Inter Congr Math, Vancouver, 1974, Vol 1. Ottawa: Canadian Math Congr, 1975, 95–118

    MATH  Google Scholar 

  25. Greco L, Iwaniec T. New inequalities for the Jacobian. Ann Inst H Poincaré Anal Non Linéaire, 1994, 11: 17–35

    Article  MathSciNet  MATH  Google Scholar 

  26. Han Y, Hofmann S. T1 theorems for Besov and Triebel-Lizorkin spaces. Trans Amer Math Soc, 1993, 337: 839–853

    MathSciNet  MATH  Google Scholar 

  27. Hofmann S. On singular integrals of Calderón-type in Rn and BMO. Rev Mat Iberoam, 1994, 10: 467–505

    Article  MATH  Google Scholar 

  28. Hofmann S. Parabolic singular integrals of Calderón-type, rough operators, and caloric layer potentials. Duke Math J, 1997, 90: 209–259

    Article  MathSciNet  MATH  Google Scholar 

  29. Lee M Y, Lin C C, Lin Y C, Yan D. Boundedness of singular integral operators with variable kernels. J Math Anal Appl, 2008, 348: 787–796

    Article  MathSciNet  MATH  Google Scholar 

  30. Meyer Y, Coifman R. Ondelettes et Opérateurs, Vol III. Paris: Hermann, 1991

    MATH  Google Scholar 

  31. Muscalu C, Schlag W. Classical and Multilinear Harmonic Analysis, Vol II. Cambridge: Cambridge Univ Press, 2013

    MATH  Google Scholar 

  32. Stein E, Weiss G. Introduction to Fourier Analysis on Euclidean Spaces. Princeton: Princeton Univ Press, 1971

    MATH  Google Scholar 

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Acknowledgements

The authors would like to express their deep gratitude to the referees for giving many valuable comments and suggestions. This work was supported by the National Natural Science Foundation of China (Grant No. 11471033), the Fundamental Research Funds for the Central Universities (Grant No. FRF-BR-17-001B), the NCET of China (Grant No. NCET-11-0574), and the Technology Plan Project of Hunan Province (Grant No. 2016TP1020).

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Authors and Affiliations

  1. Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing, 100083, China

    Yanping Chen & Liwei Wang

  2. School of Mathematics and Physics, Anhui Polytechnic University, Wuhu, 241000, China

    Liwei Wang

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  1. Yanping Chen
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  2. Liwei Wang
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Correspondence to Yanping Chen.

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Chen, Y., Wang, L. L2(ℝn) boundedness for Calderón commutator with rough variable kernel. Front. Math. China 13, 1013–1031 (2018). https://doi.org/10.1007/s11464-018-0718-8

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  • DOI: https://doi.org/10.1007/s11464-018-0718-8

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