Illinois Journal of Mathematics

Fiber-wise Calderón–Zygmund decomposition and application to a bi-dimensional paraproduct

Frédéric Bernicot

Full-text: Open access

Abstract

We are interested in a new kind of bi-dimensional bilinear paraproducts (appearing in (Amer. J. Math. 132 (2010) 201–256)), which do not fit into the setting of bilinear Calderón–Zygmund operators. In this paper, we propose a fiber-wise Calderón–Zygmund decomposition, which is specially adapted to this kind of bi-dimensional paraproduct.

Article information

Source
Illinois J. Math., Volume 56, Number 2 (2012), 415-422.

Dates
First available in Project Euclid: 22 November 2013

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1385129956

Digital Object Identifier
doi:10.1215/ijm/1385129956

Mathematical Reviews number (MathSciNet)
MR3161332

Zentralblatt MATH identifier
1290.42032

Subjects
Primary: 42B15: Multipliers

Citation

Bernicot, Frédéric. Fiber-wise Calderón–Zygmund decomposition and application to a bi-dimensional paraproduct. Illinois J. Math. 56 (2012), no. 2, 415--422. doi:10.1215/ijm/1385129956. https://projecteuclid.org/euclid.ijm/1385129956


Export citation

References

  • J. M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. Éc. Norm. Supér. 14 (1981), 209–246.
  • A. P. Calderón and A. Zygmund, On the existence of certain singular integrals, Acta Math. 88 (1952), 85–139.
  • R. R. Coifman and Y. Meyer, On commutators of singular integrals and bilinear singular integrals, Trans. Amer. Math. Soc. 212 (1975), 315–331.
  • R. R. Coifman and Y. Meyer, Commutateurs d'intégrales singulièrs et opérateurs multilinéaires, Ann. Inst. Fourier (Grenoble) 28 (1978), 177–202.
  • R. R. Coifman and Y. Meyer, Au-delà des opérateurs pseudo-diffe\'rentiels, Astérisque 57 (1978).
  • C. Demeter and C. Thiele, On the two dimensional Bilinear Hilbert Transform, Amer. J. Math. 132(1) (2010), 201–256.
  • C. Fefferman, Estimates for double Hilbert transforms, Studia Math. 44 (1972), 1–15.
  • L. Grafakos and N. Kalton, Some remarks on multilinear maps and interpolation, Math. Ann. 319 (2001), 151–180.
  • L. Grafakos and R. H. Torres, Multilinear Calderón–Zygmund theory, Adv. in Math. 165 (2002), 124–164.
  • S. Janson, On interpolation of multilinear operators, Function spaces and applications, Lecture Notes in Math., vol. 1302, Springer, Berlin, 1998, pp. 290–302.
  • C. Kenig and E. Stein, Multilinear estimates and fractional integration, Math. Res. Lett. 6 (1999), 1–15.
  • M. Lacey and C. Thiele, $L^p$ bounds for the bilinear Hilbert transform, $2<p<\infty$, Ann. of Math. 146 (1997), 693–724.
  • M. Lacey and C. Thiele, On Calderón's conjecture, Ann. of Math. 149 (1999), 475–496.
  • F. Nazarov, R. Oberlin and C. Thiele, A Calderón–Zygmund decomposition for multiple frequencies and an application to an extension of a lemma of Bourgain, Math. Res. Lett. 17(3) (2010), 529–545.