Sobolev space estimates for a class of bilinear pseudodifferential operators lacking symbolic calculus

Frederic Bernicot (1) , Rodolfo Torres (2)

1 LPP - Laboratoire Paul Painlevé - UMR 8524
2 KU - Department of Mathematiques - Kansas university

The reappearance of a 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. 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.

Mots clés: Bilinear pseudodifferential operators; exotic class; transposes; Littlewood-Paley theory; Sobolev space estimates

Domaines: Analyse classique [math.CA]

Frederic Bernicot, Rodolfo Torres. Sobolev space estimates for a class of bilinear pseudodifferential operators lacking symbolic calculus. Analysis & PDE, 2011, 4 (4), pp.551-571. ⟨hal-00468152⟩