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

Abstract

We study the bilinear Hilbert transforms and bilinear maximal functions associated to non-flat polynomial curves and obtain uniform $L^r$ estimates for $r>{d-1\over d}$, where $d$ is the degree of the polynomial. In addition, the lower bound of $r$ is sharp up to the end point.