Boundedness of bilinear multipliers whose symbols have a narrow support

TL;DR

This paper investigates the boundedness of bilinear multipliers with symbols narrowly supported around curves in the frequency plane, focusing on decay rates and geometric influences, with applications to bilinear Bochner-Riesz and restriction-extension problems.

Contribution

It provides new bounds and decay rates for bilinear multipliers with narrow support, emphasizing the role of geometric properties of the support curve.

Findings

Derived optimal decay rates depending on support width

Identified the influence of curve geometry on boundedness

Applied results to bilinear Bochner-Riesz and restriction-extension problems

Abstract

This work is devoted to studying the boundedness on Lebesgue spaces of bilinear multipliers on $\R$ whose symbol is narrowly supported around a curve (in the frequency plane). We are looking for the optimal decay rate (depending on the width of this support) for exponents satisfying a sub-H\"older scaling. As expected, the geometry of the curve plays an important role, which is described. This has applications for the bilinear Bochner-Riesz problem (in particular, boundedness of multipliers whose symbol is the characteristic function of a set), as well as for the bilinear restriction-extension problem.