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Sobolev space estimates
for a class of bilinear pseudodifferential operators lacking
symbolic calculus
Frédéric Bernicot and Rodolfo H. Torres |
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Vol. 4 (2011), No. 4, 551–571
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Abstract |
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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. |
Keywords
bilinear pseudodifferential operators, exotic class,
transposes, asymptotic expansion, elementary symbols,
Littlewood–Paley theory, Sobolev space estimates,
T(1)-Theorem
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Mathematical Subject Classification 2000
Primary: 47G30
Secondary: 42B15, 42C10, 35S99
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Milestones
Received: 23 April 2010
Revised: 2 September 2010
Accepted: 14 October 2010
Published: 9 January 2012
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Authors |
| Frédéric Bernicot | |
| Laboratoire Paul Painlevé CNRS - Université Lille 1 F-59655 Villeneuve d’Ascq, France |
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| http://math.univ-lille1.fr/~bernicot/ | |
| Rodolfo H. Torres | |
| Rodolfo H. Torres Department of Mathematics University of Kansas Lawrence, KS 66045 United States |
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| http://www.math.ku.edu/~torres/ | |
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