A uniform estimate for quartile operators

Abstract

There is a one parameter family of bilinear Hilbert transforms. Recently, some progress has been made to prove $L^p$ estimates for these operators uniformly in the parameter. In the current article we present some of these techniques in a simplified model: we prove the estimate $L^2\times L^2\to L^{1,\infty }$ uniformly for a discrete family of model operators for the bilinear Hilbert transform.