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

TL;DR

This paper investigates exotic boundedness behavior of a class of bilinear pseudodifferential operators, showing unboundedness on Lebesgue spaces but boundedness on smooth function spaces, and introduces new estimates for paramultiplication.

Contribution

It introduces a new class of bilinear pseudodifferential operators with exotic boundedness properties and develops novel estimates for a new form of paramultiplication.

Findings

Operators are unbounded on Lebesgue spaces.

Operators are bounded on spaces of smooth functions.

New estimates for a novel paramultiplication method.

Abstract

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.