Multilinear pseudodifferential operators beyond Calder\'on-Zygmund theory

TL;DR

This paper extends the theory of multilinear pseudodifferential operators by establishing boundedness results for operators with less regular symbols, beyond classical Calderón-Zygmund conditions, in weighted Lebesgue spaces.

Contribution

It proves boundedness of multilinear pseudodifferential operators with measurable symbols and investigates bilinear operators with symbols outside traditional Calderón-Zygmund classes.

Findings

Boundedness in weighted Lebesgue spaces for symbols measurable in space

New boundedness results for symbols in Hörmander classes with <1

Extension beyond classical Calderón-Zygmund theory

Abstract

We consider two types of multilinear pseudodifferential operators. First, we prove the boundedness of multilinear pseudodifferential operators with symbols which are only measurable in the spatial variables in weighted Lebesgue spaces. These results generalise earlier work of the present authors concerning linear pseudo-pseudodifferential operators. Secondly, we investigate the boundedness of bilinear pseudodifferential operators with symbols in the H\"ormander $S^{m}_{\rho, \delta}$ classes. These results are new in the case $\rho < 1$, that is, outwith the scope of multilinear Calder\'on-Zygmund theory.