The shifted bilinear Hilbert transform

TL;DR

This paper establishes $L^p$ bounds for the shifted bilinear Hilbert transform, leading to new results in bilinear ergodic averages, multiplier theorems, and singular integrals with polylogarithmic bounds.

Contribution

It provides the first $L^p$ estimates for the shifted bilinear Hilbert transform with polylogarithmic bounds, advancing the understanding of bilinear harmonic analysis.

Findings

Proved $L^p$ estimates with polylogarithmic bounds for the shifted bilinear Hilbert transform.

Derived $r$-variation estimates for bilinear ergodic averages in the sharp range $r > 2$.

Established a sharp bilinear H"ormander multiplier theorem and a $ extlog$-Dini theorem for bilinear singular integrals.

Abstract

We prove $L^p$ estimates for the shifted bilinear Hilbert transform, with a polylogarithmic bound in the size of the shift. As applications, we obtain $r$-variation estimates for bilinear ergodic averages in the sharp range $r > 2$, a sharp bilinear H\"ormander multiplier theorem, and a $\log$-Dini theorem for bilinear singular integrals.