Analysis & PDE

  • Anal. PDE
  • Volume 4, Number 4 (2011), 551-571.

Sobolev space estimates for a class of bilinear pseudodifferential operators lacking symbolic calculus

Frédéric Bernicot and Rodolfo Torres

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The reappearance of what is sometimes called exotic behavior for linear and multilinear pseudodifferential operators is investigated. The phenomenon is shown to be present in a recently introduced class of bilinear pseudodifferential operators which can be seen as more general variable coefficient counterparts of the bilinear Hilbert transform and other singular bilinear multipliers operators. We prove that such operators are unbounded on products of Lebesgue spaces but bounded on spaces of smooth functions (this is the exotic behavior referred to). In addition, by introducing a new way to approximate the product of two functions, estimates on a new paramultiplication are obtained.

Article information

Anal. PDE, Volume 4, Number 4 (2011), 551-571.

Received: 23 April 2010
Revised: 2 September 2010
Accepted: 14 October 2010
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47G30: Pseudodifferential operators [See also 35Sxx, 58Jxx]
Secondary: 42B15: Multipliers 42C10: Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) 35S99: None of the above, but in this section

bilinear pseudodifferential operators exotic class transposes asymptotic expansion elementary symbols Littlewood–Paley theory Sobolev space estimates T(1)-Theorem


Bernicot, Frédéric; Torres, Rodolfo. Sobolev space estimates for a class of bilinear pseudodifferential operators lacking symbolic calculus. Anal. PDE 4 (2011), no. 4, 551--571. doi:10.2140/apde.2011.4.551.

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  • Á. Bényi, “Bilinear pseudodifferential operators with forbidden symbols on Lipschitz and Besov spaces”, J. Math. Anal. Appl. 284:1 (2003), 97–103.
  • Á. Bényi and R. H. Torres, “Symbolic calculus and the transposes of bilinear pseudodifferential operators”, Comm. Partial Differential Equations 28:5-6 (2003), 1161–1181.
  • Á. Bényi and R. H. Torres, “Almost orthogonality and a class of bounded bilinear pseudodifferential operators”, Math. Res. Lett. 2004:1 (2004), 1–11.
  • Á. Bényi, A. R. Nahmod, and R. H. Torres, “Sobolev space estimates and symbolic calculus for bilinear pseudodifferential operators”, J. Geom. Anal. 16:3 (2006), 431–453.
  • Á. Bényi, C. Demeter, A. R. Nahmod, C. M. Thiele, R. H. Torres, and P. Villarroya, “Modulation invariant bilinear $T(1)$ theorem”, J. Anal. Math. 109 (2009), 279–352.
  • Á. Bényi, D. Maldonado, V. Naibo, and R. H. Torres, “On the Hörmander classes of bilinear pseudodifferential operators”, Integral Equations Operator Theory 67:3 (2010), 341–364.
  • F. Bernicot, “Local estimates and global continuities in Lebesgue spaces for bilinear operators”, Anal. PDE 1:1 (2008), 1–27.
  • F. Bernicot, “A bilinear pseudodifferential calculus”, J. Geom. Anal. 20:1 (2010), 39–62.
  • J.-M. Bony, “Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires”, Ann. Sci. École Norm. Sup. $(4)$ 14:2 (1981), 209–246.
  • G. Bourdaud, “Une algèbre maximale d'opérateurs pseudo-différentiels”, Comm. Partial Differential Equations 13:9 (1988), 1059–1083.
  • M. Christ and J.-L. Journé, “Polynomial growth estimates for multilinear singular integral operators”, Acta Math. 159:1-2 (1987), 51–80.
  • R. R. Coifman and Y. Meyer, “On commutators of singular integrals and bilinear singular integrals”, Trans. Amer. Math. Soc. 212 (1975), 315–331.
  • R. Coifman and Y. Meyer, “Commutateurs d'intégrales singulières et opérateurs multilinéaires”, Ann. Inst. Fourier $($Grenoble$)$ 28:3 (1978), 177–202.
  • R. R. Coifman and Y. Meyer, Au delà des opérateurs pseudo-différentiels, Astérisque 57, Société Mathématique de France, Paris, 1978.
  • G. David and J.-L. Journé, “A boundedness criterion for generalized Calderón–Zygmund operators”, Ann. of Math. $(2)$ 120:2 (1984), 371–397.
  • C. Fefferman and E. M. Stein, “Some maximal inequalities”, Amer. J. Math. 93 (1971), 107–115.
  • J. E. Gilbert and A. R. Nahmod, “Boundedness of bilinear operators with nonsmooth symbols”, Math. Res. Lett. 7:5-6 (2000), 767–778.
  • J. E. Gilbert and A. R. Nahmod, “Bilinear operators with non-smooth symbol. I”, J. Fourier Anal. Appl. 7:5 (2001), 435–467.
  • J. E. Gilbert and A. R. Nahmod, “$L\sp p$-boundedness for time-frequency paraproducts, II”, J. Fourier Anal. Appl. 8:2 (2002), 109–172.
  • L. Grafakos, Classical and modern Fourier analysis, Pearson Education, Upper Saddle River, NJ, 2004.
  • L. Grafakos and J. M. Martell, “Extrapolation of weighted norm inequalities for multivariable operators and applications”, J. Geom. Anal. 14:1 (2004), 19–46.
  • L. Grafakos and R. H. Torres, “Multilinear Calderón–Zygmund theory”, Adv. Math. 165:1 (2002), 124–164.
  • L. H örmander, “Pseudo-differential operators of type $1,1$”, Comm. Partial Differential Equations 13:9 (1988), 1085–1111.
  • L. H örmander, “Continuity of pseudo-differential operators of type $1,1$”, Comm. Partial Differential Equations 14:2 (1989), 231–243.
  • C. E. Kenig and E. M. Stein, “Multilinear estimates and fractional integration”, Math. Res. Lett. 6:1 (1999), 1–15.
  • M. Lacey and C. Thiele, “$L\sp p$ estimates on the bilinear Hilbert transform for $2<p<\infty$”, Ann. of Math. $(2)$ 146:3 (1997), 693–724.
  • M. Lacey and C. Thiele, “On Calderón's conjecture”, Ann. of Math. $(2)$ 149:2 (1999), 475–496.
  • Y. Meyer, “Régularité des solutions des équations aux dérivées partielles non linéaires (d'après J.-M. Bony)”, pp. 293–302, exposé 560 in Sém. Bourbaki, 32e année, $1979/80$, Lecture Notes in Math. 842, Springer, Berlin, 1981.
  • Y. Meyer, “Remarques sur un théorème de J.-M. Bony”, pp. 1–20 in Proceedings of the Seminar on Harmonic Analysis (Pisa, 1980), 1981. Issue as a supplement to Rend. Circ. Mat. Palermo (2).
  • C. Muscalu, T. Tao, and C. Thiele, “Multi-linear operators given by singular multipliers”, J. Amer. Math. Soc. 15:2 (2002), 469–496.
  • T. Runst, “Pseudodifferential operators of the “exotic” class $L\sp 0\sb {1,1}$ in spaces of Besov and Triebel–Lizorkin type”, Ann. Global Anal. Geom. 3:1 (1985), 13–28.
  • E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series 43, Princeton University Press, Princeton, NJ, 1993.
  • R. H. Torres, “Continuity properties of pseudodifferential operators of type $1,1$”, Comm. Partial Differential Equations 15:9 (1990), 1313–1328.
  • R. H. Torres, Boundedness results for operators with singular kernels on distribution spaces, Mem. Amer. Math. Soc. 442, American Mathematical Society, Providence, RI, 1991.
  • R. H. Torres, “Multilinear singular integral operators with variable coefficients”, Rev. Un. Mat. Argentina 50:2 (2009), 157–174.