... y)dy,Annals of Mathematics Uniform bounds for the bilinear Hilbert transforms, I By Loukas Grafakos and Xiaochun Li UNIFORM BOUNDS FOR THE BILINEAR HILBERT TRANSFORMS, I907andΦ1,k,l(·)e−2πic(ω1,k,l)(·)(x)≤ ... supported by the NSF.UNIFORM BOUNDS FOR THE BILINEAR HILBERT TRANSFORMS, I917Therefore, we obtainI132≤ C2−ηµ2−µpj|IT|12.By an argument similar to that used for the BMO estimate ... Tk= ∅ and Φj,k,T= 0. For brevity, we write(k, n) ∈ T if and only if there exists an l ∈ Zrwith (k, n, l) ∈ T.ThusUNIFORM BOUNDS FOR THE BILINEAR HILBERT TRANSFORMS, I901identifying...