Analysis & PDE
- Anal. PDE
- Volume 4, Number 4 (2011), 551-571.
Sobolev space estimates for a class of bilinear pseudodifferential operators lacking symbolic calculus
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.
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
Permanent link to this document
Digital Object Identifier
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
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