A note on bilinear multipliers with convex singularities

TL;DR

This paper investigates bounds for bilinear multipliers with convex singularities, establishing new estimates for associated paraproducts and extending results beyond the local L^2 range for specific convex curves.

Contribution

It introduces novel bounds for bilinear multipliers with convex singularities and connects these bounds to estimates for exotic paraproducts, extending the known range of applicability.

Findings

Established bounds for bilinear multipliers with convex epigraphs.

Derived estimates for exotic paraproducts using simple arguments.

Extended bounds beyond local L^2 range for polygonal convex curves.

Abstract

We study bounds in the local $L^2$ range of exponents for bilinear multipliers whose symbol is the characteristic function of the epigraph of certain convex curves. We realize these bounds as a consequence of estimates that we establish, via simple arguments, for the associated exotic paraproducts. As a further application, we observe bounds beyond the local $L^2$ range for bilinear multipliers whose symbol is the characteristic function of the epigraph of convex polygonal curves associated with these paraproducts.