Skip to main content
SpringerLink
Log in

Bilinear Square Functions and Vector-Valued Calderón-Zygmund Operators

  • Published:
Journal of Fourier Analysis and Applications Aims and scope Submit manuscript

An Erratum to this article was published on 14 December 2013

Abstract

Boundedness results for bilinear square functions and vector-valued operators on products of Lebesgue, Sobolev, and other spaces of smooth functions are presented. Bilinear vector-valued Calderón-Zygmund operators are introduced and used to obtain bounds for the optimal range of estimates in target Lebesgue spaces including exponents smaller than one.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bernicot, F.: L p estimates for non smooth bilinear Littlewood-Paley square functions on ℝ (2008). arXiv:0811.2854v1

  2. Christ, M., Journé, J.L.: Polynomial growth estimates for multilinear singular integral operators. Acta Math. 159(1–2), 51–80 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  3. David, G., Journé, J.L., Semmes, S.: Operateurs de Calderon-Zygmund, fonctions para-accretives et interpolation. Rev. Mat. Iberoam. 1(4), 1–56 (1985)

    Article  MATH  Google Scholar 

  4. Diestel, G.: Some remarks on bilinear Littlewood-Paley theory. J. Math. Anal. Appl. 307, 102–119 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Fefferman, C., Stein, E.: Some maximal inequalities. Am. J. Math. 93(1), 107–115 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  6. Grafakos, L.: Classical and Modern Fourier Analysis. Pearson/Prentice Hall, New York (2004)

    MATH  Google Scholar 

  7. Grafakos, L., Oliveira, L.: Carleson measures and associated families of multilinear operators. Preprint

  8. Grafakos, L., Torres, R.H.: On multilinear singular integrals of Calderón-Zygmund type. Publ. Mat. 57–91 (2002)

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

    Article  MathSciNet  MATH  Google Scholar 

  10. Grafakos, L., Liu, L., Maldonado, D., Yang, D.: Multilinear analysis on metric spaces. Preprint

  11. Han, Y.: Calderón-type reproducing formula and the tb theorem. Rev. Mat. Iberoam. 10(1), 51–91 (1994)

    Article  MATH  Google Scholar 

  12. Han, Y., Sawyer, E.: Littlewood-Paley theory on spaces of homogeneous type and the classical function spaces. Mem. Am. Math. Soc. 110, 530 (1994)

    MathSciNet  Google Scholar 

  13. Hofmann, S.: A local Tb theorem for square functions. Proc. Sympos. Pure Math. 175–185 (2000)

  14. Kenig, C., Stein, E.: Multilinear estimates and fractional integration. Math. Res. Lett. 6, 1–15 (1999)

    MathSciNet  MATH  Google Scholar 

  15. Lacey, M.: On bilinear Littlewood-Paley square functions. Publ. Mat. 40(2), 387–396 (1996)

    MathSciNet  MATH  Google Scholar 

  16. Maldonado, D.: Multilinear singular integrals and quadratic estimates. Doctoral Dissertation, University of Kansas (2005)

  17. Maldonado, D., Naibo, V.: On the boundedness of bilinear operators on products of Besov and Lebesgue spaces. J. Math. Anal. Appl. 352, 591–603 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Maldonado, D., Naibo, V.: Weighted norm inequalities for paraproducts and bilinear pseudodifferential operators with mild regularity. J. Fourier Anal. Appl. 15, 218–261 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Mohanty, P., Shrivastava, S.: A note on the bilinear Littlewood-Paley square function. Proc. Am. Math. Soc. 138(6), 2095–2098 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Semmes, S.: Square function estimates and the T(b) theorem. Proc. Am. Math. Soc. 110(3), 721–726 (1990)

    MathSciNet  MATH  Google Scholar 

  21. Strichartz, R.: A multilinear version of Marcinkiewicz interpolation theorem. Proc. Am. Math. Soc. 21(2), 441–444 (1969)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work is part of the author’s Ph.D. dissertation in the Mathematics Department at the University of Kansas under the supervision of Rodolfo H. Torres. The author would like to thank him for his guidance during this work. Some of the results of this work were presented in a poster at the FFT conference at the University of Maryland on February 16, 2012, and the author would like to thank the organizers of the FFT conference for the opportunity to present this work. Finally, the author would also like to thank the anonymous referees for their suggestions and comments.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Kansas, 1460 Jayhawk Blvd., Lawrence, KS, 66044, USA

    Jarod Hart

Authors
  1. Jarod Hart
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jarod Hart.

Additional information

Communicated by Pencho Petrushev.

Research supported in part by NSF Grant #DMS1069015.

An erratum to this article is available at http://dx.doi.org/10.1007/s00041-013-9314-1.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hart, J. Bilinear Square Functions and Vector-Valued Calderón-Zygmund Operators. J Fourier Anal Appl 18, 1291–1313 (2012). https://doi.org/10.1007/s00041-012-9238-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00041-012-9238-1

Keywords

Mathematics Subject Classification

Search

Navigation