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

Full-text: Open access

Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.apde/1513731168

Digital Object Identifier
doi:10.2140/apde.2011.4.551

Mathematical Reviews number (MathSciNet)
MR2872118

Zentralblatt MATH identifier
1290.47048

Subjects
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

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

Citation

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. https://projecteuclid.org/euclid.apde/1513731168


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