Sobolev norm estimates for a class of bilinear multipliers

TL;DR

This paper investigates the boundedness of specific bilinear multipliers on Sobolev spaces, extending previous L^p estimates and exploring their structural similarities to Coifman-Meyer multipliers and bilinear pseudodifferential operators.

Contribution

It provides new Sobolev space estimates for a class of bilinear multipliers related to the two-dimensional bilinear Hilbert transform, and analyzes operators with spatially dependent symbols.

Findings

Established Sobolev norm bounds for bilinear multipliers

Extended L^p estimates to Sobolev space context

Highlighted structural similarities with classical multipliers

Abstract

We consider bilinear multipliers that appeared as a distinguished particular case in the classification of two-dimensional bilinear Hilbert transforms by Demeter and Thiele [9]. In this note we investigate their boundedness on Sobolev spaces. Furthermore, we study structurally similar operators with symbols that also depend on the spatial variables. The new results build on the existing L^p estimates for a paraproduct-like operator previously studied by the authors in [5] and [10]. Our primary intention is to emphasize the analogies with Coifman-Meyer multipliers and with bilinear pseudodifferential operators of order 0.