Local estimates and global continuities in Lebesgue spaces for bilinear operators

TL;DR

This paper establishes local estimates for bilinear operators with truncated symbols and uses these to prove their global continuity in Lebesgue spaces, advancing understanding of singular bilinear operators.

Contribution

It introduces new local estimates for bilinear operators with truncated symbols and demonstrates their application to global Lebesgue space continuity.

Findings

Local estimates for bilinear operators with truncated symbols

Global Lebesgue space continuity results

Connection to bilinear Hilbert transform and singular operators

Abstract

In this paper, we first prove some local estimates for bilinear operators (closely related to the bilinear Hilbert transform and similar singular operators) with truncated symbol. Such estimates, in accordance with the Heisenberg uncertainty principle correspond to a description of ``off-diagonal'' decay. In addition they allow us to prove global continuities in Lebesgue spaces for bilinear operators with spatial dependent symbol.