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ON THE HÖRMANDER CLASSES OF BILINEAR PSEUDODIFFERENTIAL OPERATORS
, 2010
"... Bilinear pseudodifferential operators with symbols in the bilinear analog of all the Hörmander classes are considered and the possibility of a symbolic calculus for the transposes of the operators in such classes is investigated. Precise results about which classes are closed under transposition a ..."
Abstract

Cited by 10 (4 self)
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Bilinear pseudodifferential operators with symbols in the bilinear analog of all the Hörmander classes are considered and the possibility of a symbolic calculus for the transposes of the operators in such classes is investigated. Precise results about which classes are closed under transposition and can be characterized in terms of asymptotic expansions are presented. This work extends the results for more limited classes studied before in the literature and, hence, allows the use of the symbolic calculus (when it exists) as an alternative way to recover the boundedness on products of Lebesgue spaces for the classes that yield operators with bilinear CalderónZygmund kernels. Some boundedness properties for other classes with estimates in the form of Leibniz’ rule are presented as well.
Sobolev space estimates for a class of bilinear pseudodifferential operators lacking symbolic calculus
, 2010
"... The reappearance of a sometimes called exotic behavior for linear and multilinear pseudodifferential operators is investigated. The phenomenon is shown to be presentin a recently introduced class ofbilinear pseudodifferential operatorswhich can be seen as more general variable coefficient counterpa ..."
Abstract

Cited by 3 (1 self)
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The reappearance of a sometimes called exotic behavior for linear and multilinear pseudodifferential operators is investigated. The phenomenon is shown to be presentin a recently introduced class ofbilinear pseudodifferential operatorswhich can be seen as more general variable coefficient counterparts of the bilinear Hilbert transform and other singular bilinear multipliers operators. The unboundedness on product of Lebesgue spaces but the boundedness on spaces of smooth functions (which is the exotic behavior referred to) of such operators is obtained. In addition, by introducing a new way to approximate the product of two functions, estimates on a new paramultiplication are obtained.
Boundedness of smooth bilinear square functions and applications to some bilinear pseudodifferential operators
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