Skip to main content
SpringerLink
Log in

Weighted Norm Inequalities for Paraproducts and Bilinear Pseudodifferential Operators with Mild Regularity

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

Abstract

We establish boundedness properties on products of weighted Lebesgue, Hardy, and amalgam spaces of certain paraproducts and bilinear pseudodifferential operators with mild regularity. We do so by showing that these operators can be realized as generalized bilinear Calderón–Zygmund operators.

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. Bényi, Á.: Bilinear singular integrals and pseudodifferential operators. Ph.D. Thesis, University of Kansas (2002)

  2. Bényi, Á.: Bilinear pseudodifferential operators on Lipschitz and Besov spaces. J. Math. Anal. Appl. 284, 97–103 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bényi, Á., Maldonado D., Nahmod, A., Torres, R.: Bilinear paraproducts revisited. Math. Nachr. (2008, to appear)

  4. 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 

  5. Bényi, Á., Torres, R.H.: Almost orthogonality and a class of bounded bilinear pseudodifferential operators. Math. Res. Lett. 11.1, 1–12 (2004)

    Google Scholar 

  6. Bony, J.M.: Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non-linéaires. Ann. Sci. Ec. Norm. Super. Ser. 4 14(2), 209–246 (1981)

    MATH  MathSciNet  Google Scholar 

  7. Bourdaud, G.: L p-estimates for certain non-regular pseudo-differential operators. Commun. Partial Differ. Equ. 7(9), 1023–1033 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  8. Cannone, M.: Ondelettes, Paraproduits et Navier–Stokes. Diderot, Paris (1995). (French) [Wavelets, paraproducts and Navier–Stokes]

    MATH  Google Scholar 

  9. Cannone, M.: Harmonic analysis tools for solving the incompressible Navier–Stokes equations. In: Handbook of Mathematical Fluid Dynamics, vol. III, pp. 161–244. North-Holland, Amsterdam (2004)

    Google Scholar 

  10. Ching, C.-H.: Pseudo-differential operators with non-regular symbols. J. Differ. Equ. 11, 436–447 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  11. Coifman, R.R., Dobyinsky, S., Meyer, Y.: Opérateurs bilinéaires et renormalization. In: Stein, M., Fefferman, C., Fefferman, R., Wainger, S. (eds.) Essays on Fourier Analysis in Honor of Elias. Princeton University Press, Princeton (1995)

    Google Scholar 

  12. 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 

  13. Coifman, R.R., Meyer, Y.: Commutateurs d’intégrales singulières et opérateurs multilinéaires. Ann. Inst. Fourier 28(3), 177–202 (1978)

    MATH  MathSciNet  Google Scholar 

  14. Coifman, R.R., Meyer, Y.: Au-delà des Opérateurs Pseudo-Différentiels, 2nd edn. Astèrisque, vol. 57 (1978)

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

    MATH  Google Scholar 

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

    Article  Google Scholar 

  17. Fournier, J.F., Stewart, J.: Amalgams of L p and l q. Bull. Am. Math. Soc. (New Series) 13(1), 1–21 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  18. Frazier, M., Jawerth, B.: A discrete transform and decompositions of distribution spaces. J. Funct. Anal. 93, 34–169 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  19. Frazier, M., Jawerth, B., Weiss, G.: Littlewood–Paley Theory and the Study of Function Spaces. CBMS Regional Conference Series in Mathematics, vol. 79 (1991)

  20. García-Cuerva, J., Kazarian, K.: Calderón–Zygmund operators and unconditional bases of weighted Hardy spaces. Stud. Math. 109(3), 255–276 (1994)

    MATH  Google Scholar 

  21. Gilbert, J., Nahmod, A.: Bilinear operators with non-smooth symbols, I. J. Fourier Anal. Appl. 5, 435–467 (2001)

    MathSciNet  Google Scholar 

  22. Gilbert, J., Nahmod, A.: L p-boundedness of time-frequency paraproducts, II. J. Fourier Anal. Appl. 8, 109–172 (2002)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  24. Grafakos, L., Kalton, N.: The Marcinkiewicz multiplier condition for bilinear operators. Stud. Math. 146(2), 115–156 (2001)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  26. 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 

  27. Journè, J.-L.: Calderón–Zygmund Operators, Pseudo-Differential Operators and the Cauchy Integral of Calderón. Lecture Notes in Mathematics, vol. 994. Springer, Berlin (1983)

    MATH  Google Scholar 

  28. Kenig, C., Stein, E.: Multilinear estimates and fractional integration. Math. Res. Lett. 6, 1–15 (1999). Erratum in Math. Res. Lett. 6(3–4), 467 (1999)

    MATH  MathSciNet  Google Scholar 

  29. Kikuchi, N., Nakai, E., Tomita, N., Yabuta, K., Yoneda, T.: Calderón–Zygmund operators on amalgam spaces and in the discrete case. J. Math. Anal. Appl. 335, 198–212 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  30. Lacey, M.: Commutators with Riesz potentials in one and several parameters. Hokkaido Math. J. 36(1), 175–191 (2007)

    MATH  MathSciNet  Google Scholar 

  31. Lacey, M., Metcalfe, J.: Paraproducts in one and several parameters. Forum Math. 19(2), 325–351 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  32. Lannes, D.: Sharp Estimates for pseudo-differential operators with symbols of limited smoothness and commutators. J. Funct. Anal. 232, 495–539 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  33. Marschall, J.: Weighted L p-estimates for pseudo-differential operators with non-regular symbols. Z. Anal. Anwend. 10, 493–501 (1991)

    MATH  MathSciNet  Google Scholar 

  34. Marschall, J.: Non-regular pseudo-differential operators. Z. Anal. Anwend. 15, 109–148 (1996)

    MATH  MathSciNet  Google Scholar 

  35. Muscalu, C., Tao, T., Thiele, C.: Multilinear operators given by singular multipliers. J. Am. Math. Soc. 15, 469–496 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  36. Muscalu, C., Pipher, J., Tao, T., Thiele, C.: Bi-parameter paraproducts. Acta Math. 193, 269–296 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  37. Nagase, M.: The L p-boundedness of pseudo-differential equations with non-regular symbols. Commun. Partial Differ. Equ. 2(10), 1045–1061 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  38. Nishigaki, S.: Weighted norm inequalities for certain pseudo-differential operators. Tokyo J. Math. 7, 129–140 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  39. Petermichl, S.: Dyadic shifts and a logarithmic estimate for Hankel operators with matrix symbols. C. R. Acad. Sci. Paris Sèr. I Math. 330, 455–460 (2000)

    MATH  MathSciNet  Google Scholar 

  40. Sato, S.: A note on weighted estimates for certain classes of pseudo-differential operators. Rocky Mt. J. Math. 35(1), 267–284 (2005)

    Article  MATH  Google Scholar 

  41. Stefanov, A.: Pseudodifferential operators with rough symbols. Preprint

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

    MATH  Google Scholar 

  43. Taylor, M.: Tools for PDE. Pseudodifferential operators, paradifferential operators, and layer potentials. Mathematical Surveys and Monographs, vol. 81. AMS, Providence (2000)

    MATH  Google Scholar 

  44. Taylor, M.: Pseudodifferential operators and nonlinear PDE. Progress in Mathematics, vol. 100. Birkhäuser, Boston (1991)

    MATH  Google Scholar 

  45. Thiele, C.: Wave Packet Analysis. CBMS Regional Conference Series in Mathematics, vol. 105 (2006)

  46. Yabuta, K.: Generalizations of Calderón–Zygmund operators. Stud. Math. 82(1), 17–31 (1985)

    MATH  MathSciNet  Google Scholar 

  47. Yabuta, K.: Calderón–Zygmund operators and pseudodifferential operators. Commun. Partial Differ. Equ. 10(9), 1005–1022 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  48. Yabuta, K.: Weighted norm inequalities for pseudodifferential operators. Osaka J. Math. 23(3), 703–723 (1986)

    MATH  MathSciNet  Google Scholar 

  49. Youssif, A.: Bilinear operators and the Jacobian-determinant on Besov spaces. Indiana Univ. Math. J. 45, 381–396 (1996)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Kansas State University, 138 Cardwell Hall, Manhattan, KS, 66506, USA

    Diego Maldonado & Virginia Naibo

Authors
  1. Diego Maldonado
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Virginia Naibo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Virginia Naibo.

Additional information

Communicated by Hans G. Feichtinger.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Maldonado, D., Naibo, V. Weighted Norm Inequalities for Paraproducts and Bilinear Pseudodifferential Operators with Mild Regularity. J Fourier Anal Appl 15, 218–261 (2009). https://doi.org/10.1007/s00041-008-9029-x

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00041-008-9029-x

Keywords

Mathematics Subject Classification (2000)

Search

Navigation