Vol. 4, No. 4, 2011

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Sobolev space estimates for a class of bilinear pseudodifferential operators lacking symbolic calculus

Frédéric Bernicot and Rodolfo H. Torres

Vol. 4 (2011), No. 4, 551–571
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.

Keywords
bilinear pseudodifferential operators, exotic class, transposes, asymptotic expansion, elementary symbols, Littlewood–Paley theory, Sobolev space estimates, T(1)-Theorem
Mathematical Subject Classification 2000
Primary: 47G30
Secondary: 42B15, 42C10, 35S99
Milestones
Received: 23 April 2010
Revised: 2 September 2010
Accepted: 14 October 2010
Published: 9 January 2012
Authors
Frédéric Bernicot
Laboratoire Paul Painlevé
CNRS - Université Lille 1
F-59655 Villeneuve d’Ascq, France
http://math.univ-lille1.fr/~bernicot/
Rodolfo H. Torres
Rodolfo H. Torres
Department of Mathematics
University of Kansas
Lawrence, KS 66045
United States
http://www.math.ku.edu/~torres/