Revista Matemática Iberoamericana

Uniform estimates for paraproducts and related multilinear multipliers

Frédéric Bernicot

Full-text: Open access


In this paper, we prove some uniform estimates between Lebesgue and Hardy spaces for operators closely related to the multilinear paraproducts on $\mathbb{R}^d$. We are looking for uniformity with respect to parameters, which allows us to disturb the geometry and the metric on $\mathbb{R}^d$.

Article information

Rev. Mat. Iberoamericana, Volume 25, Number 3 (2009), 1055-1088.

First available in Project Euclid: 3 November 2009

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42B15: Multipliers 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B25: Maximal functions, Littlewood-Paley theory

paraproducts uniform estimate multilinear operators Littlewood-Paley theory Calderón-Zygmund decomposition


Bernicot, Frédéric. Uniform estimates for paraproducts and related multilinear multipliers. Rev. Mat. Iberoamericana 25 (2009), no. 3, 1055--1088.

Export citation


  • Bony, J. M.: Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires. Ann. Sci. École Norm. Sup. (4) 14 (1981), 209-246.
  • Bownik, M.: Boundedness of operators on Hardy spaces via atomic decompositions. Proc. Amer. Math. Soc. 133 (2005), no. 12, 3535-3542.
  • Coifman, R. and Meyer, Y.: On commutators of singular integrals and bilinear singular integrals. Trans. Amer. Math. Soc. 212 (1975), 315-331.
  • Coifman, R. and Meyer, Y.: Au delà des opérateurs pseudo-différentiels. Astérisque 57. Societé Mathématique de France, Paris, 1978.
  • Coifman, R. and Meyer, Y.: Commutateurs d'intégrales singulières et opérateurs multilinéaires. Ann. Inst. Fourier (Grenoble) 28 (1978), 177-202.
  • Coifman, R. and Meyer, Y.: Ondelettes et opérateurs III. Opérateurs multilinéaires. Actualités Mathématiques. Hermann, Paris, 1991.
  • Fan, D. and Li, X.: A bilinear oscillatory integrals along parabolas. Positivity 13 (2009), no. 2, 339-366.
  • Grafakos, L.: Classical and modern Fourier analysis. Pearson Education, Upper Saddle River, NJ, 2004.
  • Grafakos, L. and Kalton, N.: The Marcinkiewicz multiplier condition for bilinear operators. Studia Math. 146 (2001), no. 2, 115-156.
  • Grafakos, L. and Kalton, N.: Multilinear Calderón-Zygmund operators on Hardy spaces. Collect. Math. 52 (2001), 169-179.
  • Grafakos, L. and Torres, R.: Multilinear Calderón-Zygmund theory. Adv. in Math. 165 (2002), 124-164.
  • Kenig, C. and Stein, E. M.: Multilinear estimates and fractional integration. Math. Res. Lett. 6 (1999), no. 1, 1-15.
  • Li, X.: Uniform estimates for some paraproducts. New York J. Math. 14 (2008), 145-192.
  • Meda, S., Sjögren, P. and Vallarino, M.: On the $H^1$-$L^1$ boundedness of operators. Proc. Amer. Math. Soc. 136 (2008), no. 8, 2921-2931.
  • Meyer, Y., Taibleson, M. and Weiss, G.: Some functional analytic properties of the spaces $B_q$ generated by blocks. Indiana. Univ. Math. J. 34 (1985), 493-515.
  • Muscalu, C., Pipher, J., Tao, T. and Thiele, C.: A short proof of the Coifman-Meyer multilinear theorem. Non published, available at$\sim$jpipher/trilogy1.pdf.
  • Muscalu, C., Tao, T. and Thiele, C.: Uniform estimates on paraproducts. J. Anal. Math 87 (2002), 369-384.
  • Muscalu, C., Tao, T. and Thiele, C.: Uniforms estimates on multi-linear operators with modulation symmetry. J. Anal. Math 88 (2002), 255-309.
  • Stein, E. M.: Singular integrals and differentiability properties of functions. Princeton Mathematical Series 30. Princeton University Press, Princeton, N.J. 1970.
  • Stein, E. M.: Harmonic analysis: real variable methods, orthogonality, and oscillatory integrals. Princeton Mathematical Series 43. Monographs in Harmonic Analysis III. Princeton University Press, Princeton, NJ, 1993.
  • Uchiyama, A.: Hardy spaces on the Euclidean space. Springer Monographs in Mathematics. Springer-Verlag, Tokyo, 2001.