Uniform estimates for bilinear Hilbert transform and bilinear maximal functions associated to polynomials

TL;DR

This paper establishes uniform $L^r$ bounds for bilinear Hilbert transforms and maximal functions linked to polynomial curves, advancing understanding of their behavior and sharpness of estimates.

Contribution

It provides the first uniform $L^r$ estimates for these bilinear operators associated with polynomial curves, with sharp bounds up to the endpoint.

Findings

Established uniform $L^r$ estimates for $r > (d-1)/d$

Proved the sharpness of the estimates up to the endpoint

Extended the understanding of bilinear operators associated with polynomial curves

Abstract

We study the bilinear Hilbert transform and bilinear maximal functions associated to polynomial curves and obtain uniform $L^r$ estimates for $r>\frac{d-1}{d}$ and this index is sharp up to the end point.