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Smoothing of Commutators for a Hörmander Class of Bilinear Pseudodifferential Operators

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Abstract

Commutators of bilinear pseudodifferential operators with symbols in the Hörmander class \(BS_{1, 0}^{1}\) and multiplication by Lipschitz functions are shown to be bilinear Calderón-Zygmund operators. A connection with a notion of compactness in the bilinear setting for the iteration of the commutators is also made.

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References

  1. Bényi, Á., Maldonado, D., Naibo, V., Torres, R.H.: On the Hörmander classes of bilinear pseudodifferential operators. Integral Equ. Oper. Theory 67(3), 341–364 (2010)

    Article  MATH  Google Scholar 

  2. Bényi, Á., Bernicot, F., Maldonado, D., Naibo, V., Torres, R.H.: On the Hörmander classes of bilinear pseudodifferential operators, II. Indiana Univ. Math. J. To appear; Preprint at arXiv:1112.0486 [math.CA]

  3. Bényi, Á., Nahmod, A.R., Torres, R.H.: Sobolev space estimates and symbolic calculus for bilinear pseudodifferential operators. J. Geom. Anal. 16(3), 431–453 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bényi, Á., Oh, T.: On a class of bilinear pseudodifferential operators. J. Funct. Spaces Appl. 2013, 560976 (2013)

    Article  Google Scholar 

  5. Bényi, Á., Torres, R.H.: Symbolic calculus and the transposes of bilinear pseudodifferential operators. Commun. Partial Differ. Equ. 28, 1161–1181 (2003)

    Article  MATH  Google Scholar 

  6. Bényi, Á., Torres, R.H.: Compact bilinear operators and commutators. Proc. Am. Math. Soc. 141(10), 3609–3621 (2013)

    Article  MATH  Google Scholar 

  7. Bernicot, F., Maldonado, D., Moen, K., Naibo, V.: Bilinear Sobolev-Poincaré inequalities and Leibniz-type rules. J. Geom. Anal. To appear; Preprint at arXiv:1104.3942 [math.CA]

  8. Calderón, A.P.: Algebras of singular integral operators. AMS Proc. Symp. Math. 10, 18–55 (1966)

    Article  Google Scholar 

  9. Calderón, A.P.: Commutators of singular integral operators. Proc. Nat. Acad. Sci. 53, 1092–1099 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  10. Calderón, A.P.: Intermediate spaces and interpolation, the complex method. Stud. Math. 24, 113–190 (1964)

    MATH  Google Scholar 

  11. Coifman, R.R., Lions, P.L., Meyer, Y., Semmes, S.: Compensated compactness and Hardy spaces. J. Math. Pures Appl. 72, 247–286 (1993)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  13. Coifman, R.R., Meyer, Y.: Au-delà des Opérateurs Pseudo-diffeŕentiels. Astérisque, vol. 57. Société Math. de France, Paris (1978)

    MATH  Google Scholar 

  14. Coifman, R., Weiss, G.: Extension of Hardy spaces and their use in analysis. Bull. Am. Math. Soc. 83, 569–645 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  15. David, G., Journé, J.L.: A boundedness criterion for generalized Calderón-Zygmund operators. Ann. Math. 120, 371–397 (1984)

    Article  MATH  Google Scholar 

  16. Grafakos, L., Maldonado, D., Naibo, V.: A remark on an end-point Kato-Ponce inequality. Preprint at http://www.math.missouri.edu/~loukas/articles.html

  17. Grafakos, L., Oh, S.: The Kato-Ponce inequality. Commun. Partial Differ. Equs. To appear; Preprint at arXiv:1303.5144 [math.CA]

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

    Article  MATH  MathSciNet  Google Scholar 

  19. Grafakos, L., Torres, R.H.: Maximal operator and weighted norm inequalities for multilinear singular integrals. Indiana Univ. Math. J. 51(5), 1261–1276 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  20. Hart, J.: Bilinear square functions and vector-valued Calderón-Zygmund operators. J. Fourier Anal. Appl. 18(6), 1291–1313 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  21. Hart, J.: Erratum to “Bilinear square functions and vector-valued Calderón-Zygmund operators”. J. Fourier Anal. Appl. To appear; Preprint at http://www.math.wayne.edu/~jhart/MyResearch.html

  22. Hart, J.: A new proof of the bilinear T(1) theorem. Proc. Am. Math. Soc. To appear; Preprint at http://www.math.wayne.edu/~jhart/MyResearch.html

  23. Hart, J.: A bilinear T(b) theorem for singular integral operators. Preprint at arXiv:1306.0385 [math.CA]

  24. Iwaniec, T.: Nonlinear commutators and Jacobians. J. Fourier Anal. Appl. 3, 775–796 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  25. Iwaniec, T., Sbordone, C.: Riesz transform and elliptic PDEs with VMO coefficients. J. Anal. Math. 74, 183–212 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  26. Kato, T., Ponce, G.: Commutator estimates and the Euler and Navier-Stokes equations. Commun. Pure Appl. Math. 41, 891–907 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  27. Kenig, C., Ponce, G., Vega, L.: On unique continuation for nonlinear Schrödinger equations. Commun. Pure Appl. Math. 56, 1247–1262 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  28. Meyer, Y., Coifman, R.R.: Wavelets: Calderón-Zygmund and Multilinear Operators. Cambridge University Press, Cambridge (1997)

    MATH  Google Scholar 

  29. Miyachi, A., Tomita, N.: Calderón-Vaillancourt type theorem for bilinear pseudo-differential operators. Indiana Univ. Math. J. To appear

  30. Pérez, C., Torres, R.H.: Sharp maximal function estimates for multilinear singular integrals. Contemp. Math. 320, 323–331 (2003)

    Article  Google Scholar 

  31. Pérez, C., Pradolini, G., Torres, R.H., Trujillo-González, R.: End-points estimates for iterated commutators of multilinear singular integrals. Bull. Lond. Math. Soc. To appear; Preprint at arXiv:1004.4976 [math.CA]

  32. Stein, E.: Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton (1993), 695 pp.

    MATH  Google Scholar 

  33. Tang, L.: Weighted estimates for vector-valued commutators of multilinear operators. Proc. R. Soc. Edinb., Sect. A, Math. 138, 897–922 (2008)

    Article  MATH  Google Scholar 

  34. Uchiyama, A.: On the compactness of operators of Hankel type. Tôhoku Math. J. 30(1), 163–171 (1978)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Western Washington University, 516 High St, Bellingham, WA, 98225, USA

    Árpád Bényi

  2. School of Mathematics, The University of Edinburgh and The Maxwell Institute for the Mathematical Sciences, James Clerk Maxwell Building, The King’s Buildings, Mayfield Road, Edinburgh, EH9 3JZ, Scotland

    Tadahiro Oh

Authors
  1. Árpád Bényi
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  2. Tadahiro Oh
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Correspondence to Árpád Bényi.

Additional information

Communicated by Loukas Grafakos.

This work is partially supported by a grant from the Simons Foundation Grant (No. 246024 to Árpád Bényi). The second author acknowledges support from an AMS-Simons Travel Grant.

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Bényi, Á., Oh, T. Smoothing of Commutators for a Hörmander Class of Bilinear Pseudodifferential Operators. J Fourier Anal Appl 20, 282–300 (2014). https://doi.org/10.1007/s00041-013-9312-3

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  • DOI: https://doi.org/10.1007/s00041-013-9312-3

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